Include the names of your collaborators here.
This homework assignment reviews important aspects of training linear models focusing on the relationship between complexity and model performance. You will fit non-Bayesian and Bayesian linear models of varying levels of complexity. You will compare their performance and examine their predictive trends. Unlike earlier assignments you will examine both types of uncertainty in addition to the trend. This way you gain experience with the relationship between complexity, training set performance, confidence intervals, and prediction intervals.
This assignment also requires you to fit the Bayesian models with various prior standard deviations. In this way, you learn about the prior’s role on coefficient estimates and predictive trends. These tasks introduce to the concept that the prior regularizes coefficients.
Lastly, you are introduced to non-Bayesian regularization with Lasso
regression via the glmnet package. If you do not have
glmnet installed PLEASE download it before
starting the assignment.
IMPORTANT: The RMarkdown assumes you have downloaded the data set (CSV file) to the same directory you saved the template Rmarkdown file. If you do not have the CSV files in the correct location, the data will not be loaded correctly.
Certain code chunks are created for you. Each code chunk has
eval=TRUE set in the chunk options. You
MUST change it to be eval=TRUE in order
for the code chunks to be evaluated when rendering the document.
You are free to add more code chunks if you would like.
This assignment will use packages from the tidyverse
suite as well as the coefplot package. Those packages are
imported for you below.
library(tidyverse)
## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr 1.1.2 ✔ readr 2.1.4
## ✔ forcats 1.0.0 ✔ stringr 1.5.0
## ✔ ggplot2 3.4.3 ✔ tibble 3.2.1
## ✔ lubridate 1.9.2 ✔ tidyr 1.3.0
## ✔ purrr 1.0.2
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
library(coefplot)
This assignment also uses the splines and
MASS packages. Both are installed with base R
and so you do not need to download any additional packages to complete
the assignment.
The last question in the assignment uses the glmnet
package. As stated previously, please download and install
glmnet if you do not currently have it.
You will fit and compare 6 models of varying
complexity using non-Bayesian methods. The unknown
parameters will be be estimated by finding their Maximum Likelihood
Estimates (MLE). You are allowed to use the lm() function
for this problem.
The data are loaded in the code chunk and a glimpse is shown for you
below. There are 2 continuous inputs, x1 and
x2, and a continuous response y.
hw_file_path <- 'hw08_data.csv'
df <- readr::read_csv(hw_file_path, col_names = TRUE)
## Rows: 100 Columns: 3
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (3): x1, x2, y
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
df %>% glimpse()
## Rows: 100
## Columns: 3
## $ x1 <dbl> -0.30923281, 0.63127211, -0.68276690, 0.26930562, 0.37252021, 1.296…
## $ x2 <dbl> 0.308779853, -0.547919793, 2.166449412, 1.209703658, 0.785485991, -…
## $ y <dbl> 0.43636596, 1.37562976, -0.84366730, -0.43080811, 0.77456951, 1.361…
Create a scatter plot between the response, y,
and each input using ggplot().
Based on the visualizations, do you think there are trends between either input and the response?
df %>%
ggplot(mapping = aes(y = y)) +
geom_point(mapping = aes(x = x1, color = 'red')) +
geom_point(mapping = aes(x = x2, color = 'blue'))
You will fit multiple models of varying complexity in this problem. You will start with linear additive features which add the effect of one input with the other. Your model therefore controls for both inputs.
Fit a model with linear additive features to predict the
response, y. Use the formula interface and the
lm() function to fit the model. Assign the result to the
mod01 object.
Visualize the coefficient summaries with the
coefplot() function. Are any of the features statistically
significant?
### add more code chunks if you like
mod01 <- lm(y ~ x1 + x2, data = df)
coefplot(mod01)
As discussed in lecture, we can derive features from inputs. We have worked with polynomial features and spline-based features in previous assignments. Features can also be derived as the products between different inputs. A feature calculated as the product of multiple inputs is usually referred to as the interaction between those inputs.
In the formula interface, a product of two inputs is denoted by the
:. And so if we want to include just the multiplication of
x1 and x2 in a model we would type,
x1:x2. We can then include main-effect
terms by including the additive features within the formula. Thus, the
formula for a model with additive features and the interaction between
x1 and x2 is:
y ~ x1 + x2 + x1:x2
However, the formula interface provides a short-cut to create main
effects and interaction features. In the formula interface, the
* operator will generate all main-effects and all
interactions for us.
Fit a model with all main-effect and all-interaction features
between x1 and x2 using the short-cut
* operator within the formula interface. Assign the result
to the mod02 object.
Visualize the coefficient summaries with the
coefplot() function. How many features are present in the
model? Are any of the features statistically significant?
### add more code chunks if you like
mod02 <- lm(y ~ x1*x2, data = df)
coefplot(mod02)
The * operator will interact more than just inputs. We
can interact expressions or groups of features together. To interact one
group of features by another group of features, we just need to enclose
each group within parenthesis, (), and separate them by the
* operator. The line of code below shows how this works
with the <expression 1> and
<expression 2> as place holders for any expression we
want to use.
(<expression 1>) * (<expression 2>)
Fit a model which interacts linear and quadratic features
from x1 with linear and quadratic features from
x2. Assign the result to the mod03
object.
Visualize the coefficient summaries with the
coefplot() function. How many features are present in the
model? Are any of the features statistically significant?
HINT: Remember to use the I() function when
typing polynomials in the formula interface.
### add more code chunks if you like
mod03 <- lm(y ~ (x1 + I(x1^2))*(x2 + I(x2^2)), data = df)
coefplot(mod03)
Let’s now try a more complicated model.
Fit a model which interacts linear, quadratic, cubic, and
quartic (4th degree) polynomial features from x1 with
linear, quadratic, cubic, and quartic (4th degree) polynomial features
from x2. Assign the result to the mod04
object.
Visualize the coefficient summaries with the
coefplot() function. Are any of the features statistically
significant?
### add more code chunks if you like
mod04 <- lm(y ~ (x1 + I(x1^2) + I(x1^3) + I(x1^4)) * (x2 + I(x2^2) + I(x2^3) + I(x2^4)), data = df)
coefplot(mod04)
Let’s try using spline based features. We will use a high
degree-of-freedom natural spline applied to x1 and interact
those features with polynomial features derived from
x2.
Fit a model which interacts a 12 degree-of-freedom natural
(DOF) spline from x1 with linear and quadrtic polyonomial
features from x2. Assign the result to
mod05.
Visualize the coefficient summaries with the
coefplot() function. Are any of the features statistically
significant?
### add more code chunks if you like
mod05 <- lm(y ~ splines::ns(x1, df = 12) * I(x2^2), data = df)
coefplot(mod05)
Let’s fit one final model.
Fit a model which interacts a 12 degree-of-freedom natural
spline from x1 with linear, quadrtic, cubic, and quartic
(4th degree) polyonomial features from x2. Assign the
result to mod05.
Visualize the coefficient summaries with the
coefplot() function. Are any of the features statistically
significant?
### add more code chunks if you like
mod06 <- lm(y ~ splines::ns(x1, df = 12) * (x2 +I(x2^2) + I(x2^3) + I(x2^4)), data = df)
coefplot(mod06)
Now that you have fit multiple models of varying complexity, it is time to identify the best performing model.
Identify the best model considering training set only performance metrics. Which model is best according to R-squared? Which model is best according to AIC? Which model is best according to BIC?
HINT: The brooom::glance() function can be
helpful here. The broom package is installed with
tidyverse and so you should have it already.
### add more code chunks if you like
models_glance <- bind_rows(
broom::glance(mod01),
broom::glance(mod02),
broom::glance(mod03),
broom::glance(mod04),
broom::glance(mod05),
broom::glance(mod06),
)
models_glance
## # A tibble: 6 × 12
## r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.0594 0.0401 0.980 3.07 5.12e- 2 2 -138. 285. 295.
## 2 0.113 0.0853 0.956 4.08 9.00e- 3 3 -135. 281. 294.
## 3 0.547 0.507 0.702 13.7 7.25e-13 8 -102. 224. 250.
## 4 0.599 0.470 0.728 4.66 1.51e- 7 24 -95.7 243. 311.
## 5 0.647 0.527 0.688 5.42 7.03e- 9 25 -89.4 233. 303.
## 6 0.782 0.383 0.785 1.96 1.64e- 2 64 -65.3 263. 434.
## # ℹ 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
sprintf('mod0%d', which.max(models_glance$r.squared))
## [1] "mod06"
sprintf('mod0%d', which.min(models_glance$AIC))
## [1] "mod03"
Now that you know which model is best, let’s visualize the predictive trends from the six models. This will help us better understand their performance and behavior.
You will define a prediction or visualization test grid. This grid
will allow you to visualize behavior with respect to x1 for
multiple values of x2.
Create a grid of input values where x1 consists
of 101 evenly spaced points between -3.2 and 3.2 and x2 is
9 evenly spaced points between -3 and 3. The expand.grid()
function is started for you and the data type conversion is provided to
force the result to be a tibble.
?seq
viz_grid <- expand.grid(x1 = seq(-3.2, 3.2, length.out = 101),
x2 = seq(-3, 3, length.out = 9),
KEEP.OUT.ATTRS = FALSE,
stringsAsFactors = FALSE) %>%
as.data.frame() %>% tibble::as_tibble()
You will make predictions for each of the models and visualize their
trends. A function, tidy_predict(), is created for you
which assembles the predicted mean trend, the confidence interval, and
the prediction interval into a tibble for you. The result
include the input values to streamline making the visualizations.
tidy_predict <- function(mod, xnew)
{
pred_df <- predict(mod, xnew, interval = "confidence") %>%
as.data.frame() %>% tibble::as_tibble() %>%
dplyr::select(pred = fit, ci_lwr = lwr, ci_upr = upr) %>%
bind_cols(predict(mod, xnew, interval = 'prediction') %>%
as.data.frame() %>% tibble::as_tibble() %>%
dplyr::select(pred_lwr = lwr, pred_upr = upr))
xnew %>% bind_cols(pred_df)
}
The first argument to the tidy_predict() function is a
lm() model object and the second argument is new or test
dataframe of inputs. When working with lm() and its
predict() method, the functions will create the test design
matrix consistent with the training design basis. It does so via the
model object’s formula which is contained within the lm()
model object. The lm() object therefore takes care of the
heavy lifting for us!
Make predictions with each of the six models you fit in
Problem 01 using the visualization grid, viz_grid. The
predictions should be assigned to the variables pred_lm_01
through pred_lm_06 where the number is consistent with the
model number fit previously.
pred_lm_01 <- tidy_predict(mod01, viz_grid)
pred_lm_02 <- tidy_predict(mod02, viz_grid)
pred_lm_03 <- tidy_predict(mod03, viz_grid)
pred_lm_04 <- tidy_predict(mod04, viz_grid)
pred_lm_05 <- tidy_predict(mod05, viz_grid)
pred_lm_06 <- tidy_predict(mod06, viz_grid)
You will now visualize the predictive trends and the confidence and
prediction intervals for each model. The pred column in of
each pred_lm_ objects is the predictive mean trend. The
ci_lwr and ci_upr columns are the lower and
upper bounds of the confidence interval, respectively. The
pred_lwr and pred_upr columns are the lower
and upper bounds of the prediction interval, respectively.
You will use ggplot() to visualize the predictions. You
will use geom_line() to visualize the mean trend and
geom_ribbon() to visualize the uncertainty intervals.
Visualize the predictions of each model on the visualization
grid. Pipe the pred_lm_ object to ggplot() and
map the x1 variable to the x-aesthetic. Add three geometric
object layers. The first and second layers are each
geom_ribbon() and the third layer is
geom_line(). In the geom_line() layer map the
pred variable to the y aesthetic. In the first
geom_ribbon() layer, map pred_lwr and
pred_upr to the ymin and ymax
aesthetics, respectively. Hard code the fill to be orange
in the first geom_ribbon() layer (outside the
aes() call). In the second geom_ribbon()
layer, map ci_lwr and ci_upr to the
ymin and ymax aesthetics, respectively. Hard
code the fill to be grey in the second
geom_ribbon() layer (outside the aes() call).
Include facet_wrap() with the facets with controlled by the
x2 variable.
To help compare the visualizations across models include a
coord_cartesian() layer with the ylim argument
set to c(-7,7).
Each model’s prediction visualization should be created in a separate code chunk.
Create separate code chunks for each visualization.
pred_lm_01 %>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr), fill = 'orange') +
geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
geom_line(mapping = aes(y = pred)) +
facet_wrap(~ x2)
pred_lm_02 %>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr), fill = 'orange') +
geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
geom_line(mapping = aes(y = pred)) +
facet_wrap(~ x2)
pred_lm_03 %>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr), fill = 'orange') +
geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
geom_line(mapping = aes(y = pred)) +
facet_wrap(~ x2)
pred_lm_04 %>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr), fill = 'orange') +
geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
geom_line(mapping = aes(y = pred)) +
facet_wrap(~ x2)
pred_lm_05 %>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr), fill = 'orange') +
geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
geom_line(mapping = aes(y = pred)) +
facet_wrap(~ x2)
pred_lm_06 %>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = pred_lwr, ymax = pred_upr), fill = 'orange') +
geom_ribbon(mapping = aes(ymin = ci_lwr, ymax = ci_upr), fill = 'grey') +
geom_line(mapping = aes(y = pred)) +
facet_wrap(~ x2)
Do you feel the predictions are consistent with the model performance rankings based on AIC/BIC? What is the defining characteristic of the models considered to be the worst by AIC/BIC?
What do you think?
AIC considers the number of parameters as a penalty. Yes, I feel the rankings according to AIC are justified because the we see the scale of confidence interval is very high for the complex model, which means that the models have no idea where the answer lies.
Now that you have fit non-Bayesian linear models with maximum likelihood estimation, it is time to use Bayesian models to understand the influence of the prior on the model behavior.
Regardless of your answers in Problem 02 you will only work with model 3 and model 6 in this problem.
You will perform the Bayesian analysis using the Laplace Approximation just as you did in the previous assignment. You will define the log-posterior function just as you did in the previous assignment and so before doing so you must create the list of required information. This list will include the observed response, the design matrix, and the prior specification. You will use independent Gaussian priors on the regression parameters with a shared prior mean and shared prior standard deviation. You will use an Exponential prior on the unknown likelihood noise (the \(\sigma\) parameter).
Complete the two code chunks below. In the first, create the
design matrix following mod03’s formula, and assign the
object to the X03 variable. Complete the
info_03_weak list by assigning the response to
yobs and the design matrix to design_matrix.
Specify the shared prior mean, mu_beta, to be 0, the shared
prior standard deviation, tau_beta, as 50, and the rate
parameter on the noise, sigma_rate, to be 1.
Complete the second code chunk with the same prior
specification. The second code chunk however requires that you create
the design matrix associated with mod06’s formula and
assign the object to the X06 variable. Assign
X06 to the design_matrix field of the
info_06_weak list.
X03 <- model.matrix(y ~ (x1 + I(x1^2))*(x2 + I(x2^2)), data = df)
info_03_weak <- list(
yobs = df$y,
design_matrix = X03,
mu_beta = 0,
tau_beta = 50,
sigma_rate = 1
)
X06 <- model.matrix(y ~ splines::ns(x1, df = 12) * (x2 +I(x2^2) + I(x2^3) + I(x2^4)), data = df)
info_06_weak <- list(
yobs = df$y,
design_matrix = X06,
mu_beta = 0,
tau_beta = 50,
sigma_rate = 1
)
You will now define the log-posterior function
lm_logpost(). You will continue to use the
log-transformation on \(\sigma\), and
so you will actually define the log-posterior in terms of the mean trend
\(\boldsymbol{\beta}\)-parameters and
the unbounded noise parameter, \(\varphi =
\log\left[\sigma\right]\).
The comments in the code chunk below tell you what you need to fill
in. The unknown parameters to learn are contained within the first input
argument, unknowns. You will assume that the unknown \(\boldsymbol{\beta}\)-parameters are listed
before the unknown \(\varphi\)
parameter in the unknowns vector. You must specify the
number of \(\boldsymbol{\beta}\)
parameters programmatically to allow scaling up your function to an
arbitrary number of unknowns. You will assume that all variables
contained in the my_info list (the second argument to
lm_logpost()) are the same fields in the
info_03_weak list you defined in Problem 3a).
Define the log-posterior function by completing the code
chunk below. You must calculate the mean trend, mu, using
matrix math between the design matrix and the unknown \(\boldsymbol{\beta}\) column vector.
HINT: This function should look very famaliar…
lm_logpost <- function(unknowns, my_info)
{
length_beta <- length(unknowns) - 1
# extract the beta parameters from the `unknowns` vector
beta_v <- unknowns[1:length_beta] %>% as.matrix()
# extract the unbounded noise parameter, varphi
lik_varphi <- unknowns[length(unknowns)]
# back-transform from varphi to sigma
lik_sigma <- exp(lik_varphi)
# extract design matrix
X <- my_info$design_matrix
# calculate the linear predictor
mu <- X %*% beta_v
# evaluate the log-likelihood
log_lik <- sum(dnorm(my_info$yobs, mean = mu, sd = lik_sigma, log = TRUE))
# evaluate the log-prior
log_prior_beta <- sum(dnorm(beta_v, mean = my_info$mu_beta, sd = my_info$tau_beta, log = TRUE))
log_prior_sigma <- dexp(lik_sigma, rate = my_info$sigma_rate, log = TRUE)
# add the mean trend prior and noise prior together
log_prior <- log_prior_beta + log_prior_sigma
# account for the transformation
log_derive_adjust <- lik_varphi
# sum together
log_lik + log_prior + lik_varphi
}
The my_laplace() function is defined for you in the code
chunk below. This function executes the laplace approximation and
returns the object consisting of the posterior mode, posterior
covariance matrix, and the log-evidence.
my_laplace <- function(start_guess, logpost_func, ...)
{
# code adapted from the `LearnBayes`` function `laplace()`
fit <- optim(start_guess,
logpost_func,
gr = NULL,
...,
method = "BFGS",
hessian = TRUE,
control = list(fnscale = -1, maxit = 1001))
mode <- fit$par
post_var_matrix <- -solve(fit$hessian)
p <- length(mode)
int <- p/2 * log(2 * pi) + 0.5 * log(det(post_var_matrix)) + logpost_func(mode, ...)
# package all of the results into a list
list(mode = mode,
var_matrix = post_var_matrix,
log_evidence = int,
converge = ifelse(fit$convergence == 0,
"YES",
"NO"),
iter_counts = as.numeric(fit$counts[1]))
}
Execute the Laplace Approximation for the model 3 formulation
and the model 6 formulation. Assign the model 3 result to the
laplace_03_weak object, and assign the model 6 result to
the laplace_06_weak object. Check that the optimization
scheme converged.
start_guess_03 <- c(rnorm(info_03_weak$design_matrix %>% ncol()), 5)
start_guess_06 <- c(rnorm(info_06_weak$design_matrix %>% ncol()), 5)
### add more code chunks if you like
laplace_03_weak <- my_laplace(start_guess_03,
lm_logpost,
info_03_weak)
laplace_03_weak
## $mode
## [1] 0.665292190 0.164471932 -0.160370454 -0.051975252 -0.556456896
## [6] 0.122792393 -0.082521271 0.005338202 0.020383961 -0.398805777
##
## $var_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 1.382439e-02 -2.857431e-04 -6.974406e-03 3.995513e-04 -5.817154e-03
## [2,] -2.857431e-04 7.785559e-03 1.697286e-04 8.041367e-04 -5.563797e-04
## [3,] -6.974406e-03 1.697286e-04 7.581133e-03 1.620084e-04 3.406225e-03
## [4,] 3.995513e-04 8.041367e-04 1.620084e-04 8.160218e-03 5.829452e-04
## [5,] -5.817154e-03 -5.563797e-04 3.406225e-03 5.829452e-04 5.670717e-03
## [6,] 2.435675e-03 -2.044357e-05 1.411762e-04 6.038636e-04 -1.933566e-03
## [7,] -1.669154e-03 -2.966175e-03 6.048780e-04 -1.077101e-03 1.525058e-03
## [8,] 1.328242e-03 1.841321e-04 -5.911626e-04 -2.460648e-03 -1.598428e-03
## [9,] 2.527204e-03 3.147396e-04 -3.492292e-03 -9.078418e-04 -2.395753e-03
## [10,] -6.188075e-08 -8.510241e-09 5.193266e-08 4.787207e-08 5.758501e-08
## [,6] [,7] [,8] [,9] [,10]
## [1,] 2.435675e-03 -1.669154e-03 1.328242e-03 2.527204e-03 -6.188075e-08
## [2,] -2.044357e-05 -2.966175e-03 1.841321e-04 3.147396e-04 -8.510241e-09
## [3,] 1.411762e-04 6.048780e-04 -5.911626e-04 -3.492292e-03 5.193266e-08
## [4,] 6.038636e-04 -1.077101e-03 -2.460648e-03 -9.078418e-04 4.787207e-08
## [5,] -1.933566e-03 1.525058e-03 -1.598428e-03 -2.395753e-03 5.758501e-08
## [6,] 1.216969e-02 -4.522204e-03 4.252484e-03 -2.772279e-03 1.666019e-08
## [7,] -4.522204e-03 4.978960e-03 -2.548154e-03 8.311601e-04 -6.159891e-09
## [8,] 4.252484e-03 -2.548154e-03 4.465035e-03 -4.413527e-04 -2.726561e-08
## [9,] -2.772279e-03 8.311601e-04 -4.413527e-04 3.461300e-03 -3.111312e-08
## [10,] 1.666019e-08 -6.159891e-09 -2.726561e-08 -3.111312e-08 4.999649e-03
##
## $log_evidence
## [1] -164.7934
##
## $converge
## [1] "YES"
##
## $iter_counts
## [1] 123
laplace_06_weak <- my_laplace(start_guess_06,
lm_logpost,
info_06_weak)
laplace_06_weak
## $mode
## [1] -6.6484595 3.4962633 9.0989632 4.5605887 9.0125568 6.8649271
## [7] 7.6500717 7.3278346 7.5069870 4.8386197 4.8822841 16.7361496
## [13] 2.3932857 1.4046485 7.3245263 -1.5132287 -2.8215250 -0.4551382
## [19] -3.9582350 1.6387343 -2.7248122 0.2896645 -1.8434055 -1.9327134
## [25] -3.5033200 -0.1594400 -5.7746471 3.5040164 12.9820301 1.5039461
## [31] -16.5476990 3.6503851 -12.5042022 -4.8727732 -9.8597094 -10.4641293
## [37] -7.4626111 10.8662320 -11.8003938 -21.7176819 0.9385730 -1.1289199
## [43] 3.1357242 -2.0794339 2.9141041 -0.1445471 2.4543364 2.7035451
## [49] -4.3120943 17.7212086 -4.8113605 -22.8717162 -39.0220452 0.3723432
## [55] 4.9885424 0.1148528 3.6193377 2.4902689 2.2639822 7.1160764
## [61] -7.4197172 -7.1080440 4.3518312 21.8012963 23.0560070 -0.7575425
##
## $var_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 13.955694350 -11.102762573 -15.43148649 -12.797002304 -14.648632092
## [2,] -11.102762573 10.194689588 11.51446999 10.733707353 11.348153488
## [3,] -15.431486493 11.514469987 17.66663342 13.678032610 16.454327361
## [4,] -12.797002304 10.733707353 13.67803261 12.550161967 12.990218706
## [5,] -14.648632092 11.348153488 16.45432736 12.990218706 15.825030183
## [6,] -13.589283709 10.994001775 14.86643111 12.792036757 13.921181271
## [7,] -14.172633004 11.149071909 15.78293221 12.764366104 15.099903083
## [8,] -13.866391483 11.092946474 15.27971403 12.823468623 14.445503330
## [9,] -14.034541892 11.134143952 15.54644299 12.812939782 14.785920834
## [10,] -13.778910688 10.998562656 15.20639531 12.675200993 14.428941274
## [11,] -7.314751552 6.219533640 7.86703853 6.853745311 7.608102257
## [12,] -30.182786102 23.004292381 33.94082552 27.281567678 31.888884233
## [13,] -3.976483671 3.790625345 4.04423277 3.903597866 4.011779850
## [14,] -17.616397897 17.059021874 17.69271231 16.900366488 18.078816545
## [15,] -5.451030165 1.273024871 7.68713039 4.205107484 6.157389308
## [16,] 9.909650357 -9.848645254 -9.77248337 -9.635941920 -10.106471751
## [17,] -0.759044598 2.008961109 0.11194327 1.013471878 0.617044419
## [18,] 17.063250805 -16.643762681 -17.06211592 -16.480372164 -17.475822229
## [19,] 18.063796219 -17.166866963 -18.43731827 -16.896244906 -18.672580838
## [20,] 16.901093565 -16.938914984 -16.45365961 -17.289458026 -16.941577154
## [21,] 17.804119225 -17.047929934 -18.06328437 -16.691138698 -18.419997152
## [22,] 17.514884801 -17.124664139 -17.42418007 -17.235694535 -17.704477340
## [23,] 18.024551420 -17.135342703 -18.36948310 -16.834072800 -18.819740993
## [24,] 17.612128673 -17.048486171 -17.69045433 -16.893588006 -18.118147294
## [25,] 17.581559500 -17.067477476 -17.62464316 -16.927333537 -17.961785612
## [26,] 17.662533040 -17.036260844 -17.78288156 -16.911162072 -18.178700298
## [27,] 10.553336000 -9.978989839 -10.74470517 -10.028054015 -10.864855703
## [28,] 34.933780037 -34.394389140 -34.75630821 -33.704511776 -35.746009554
## [29,] 5.639526332 -5.499723543 -5.57407574 -5.585278488 -5.742769766
## [30,] 0.063331179 -0.849283800 0.36538347 -0.994651535 0.438597799
## [31,] 8.040312116 -1.016040777 -12.48538643 -4.594179368 -9.810307699
## [32,] 2.168369387 -0.647248240 -2.29934532 -4.355221411 -1.274519786
## [33,] 7.209262498 -1.768795725 -10.41560810 -4.429895079 -9.123874148
## [34,] 4.936019159 -1.280269144 -6.71212547 -4.761315593 -4.737023137
## [35,] 5.568448278 -1.163908466 -8.02975996 -3.680583456 -6.810882439
## [36,] 5.363801199 -1.393234133 -7.41545692 -4.530640311 -5.711911985
## [37,] 5.547885005 -1.243484364 -7.90240172 -4.022686727 -6.485801071
## [38,] 4.369824271 -0.585393445 -6.32934695 -3.452734126 -4.896588180
## [39,] 1.870256569 -0.970715789 -2.37628965 -1.710420735 -1.968293538
## [40,] 14.411171048 -2.091994209 -21.00923451 -10.301396199 -16.687025846
## [41,] -2.866588883 1.495017343 3.62797769 1.988858723 3.407166161
## [42,] -7.615807691 8.629924812 6.98914328 7.822602615 7.535635558
## [43,] -10.814637403 10.351901788 10.87800005 10.229894577 11.154313171
## [44,] -8.753051725 9.510321748 7.98796127 9.483323036 8.497264834
## [45,] -10.432200160 10.004887147 10.57327291 9.739540568 10.862393981
## [46,] -9.703408430 9.853845996 9.38249886 9.841150835 9.593547248
## [47,] -10.344544250 9.945948486 10.47058831 9.631372289 10.878646662
## [48,] -10.019356690 9.858608379 9.96507361 9.583136248 10.373082811
## [49,] -9.388086067 9.594583538 9.03957676 9.479034348 9.289797789
## [50,] -10.733366051 10.301237754 10.85678663 10.169844010 11.119702623
## [51,] -4.468201765 4.385318660 4.41655054 4.380063461 4.516162380
## [52,] -19.665507969 20.761224845 18.58945994 19.760807857 19.727293258
## [53,] 4.360412067 -2.972063169 -5.30274767 -3.287060437 -4.903149591
## [54,] 1.534652822 -1.452967358 -1.59693282 -1.225519804 -1.701071509
## [55,] 0.360873971 -2.305200574 0.83290584 -1.198733960 0.049516742
## [56,] 1.417049892 -2.046856206 -1.28501216 -0.690686668 -1.682978780
## [57,] 0.470552673 -1.965057063 0.37743091 -1.120404069 -0.039781553
## [58,] 0.496015366 -1.812761275 0.15829432 -0.502215393 -0.576729193
## [59,] 1.160449771 -2.238805588 -0.60671650 -1.487156753 -0.941314020
## [60,] 0.545447340 -1.775160767 0.06599072 -0.653187576 -0.557448131
## [61,] 1.398485892 -2.519315506 -0.82006357 -1.664448300 -1.178312474
## [62,] 1.210464137 -2.254010473 -0.70983594 -1.298327996 -1.172483669
## [63,] 0.319353424 -0.419379264 -0.27141649 -0.256187050 -0.340418547
## [64,] 0.297834316 -4.852509384 2.15412232 -1.726512965 0.473911765
## [65,] -2.809501119 2.523497171 3.03638984 2.492064039 2.953538562
## [66,] 0.008110337 -0.003305448 -0.01084178 -0.006145322 -0.009196886
## [,6] [,7] [,8] [,9] [,10]
## [1,] -13.589283709 -14.172633004 -13.866391483 -14.034541892 -13.778910688
## [2,] 10.994001774 11.149071909 11.092946474 11.134143952 10.998562656
## [3,] 14.866431108 15.782932208 15.279714029 15.546442995 15.206395314
## [4,] 12.792036757 12.764366104 12.823468623 12.812939781 12.675200993
## [5,] 13.921181271 15.099903083 14.445503330 14.785920834 14.428941274
## [6,] 13.778726255 13.405693134 13.725913464 13.552716596 13.478330072
## [7,] 13.405693134 15.295689075 13.402286447 14.608360142 13.806778925
## [8,] 13.725913464 13.402286447 14.649169506 13.473326522 13.925288215
## [9,] 13.552716596 14.608360142 13.473326522 14.742673349 13.395484526
## [10,] 13.478330072 13.806778925 13.925288215 13.395484526 14.897966476
## [11,] 7.137587705 7.516957025 7.117039760 7.691306627 6.371284356
## [12,] 29.282913514 30.653990786 30.071761104 30.131491829 30.262614703
## [13,] 4.026542496 3.735043649 4.318836136 3.326129647 5.374069739
## [14,] 17.505622463 17.642375355 17.672698780 17.672774763 17.379336876
## [15,] 4.955870211 5.782115558 5.257458613 5.531680579 5.356681646
## [16,] -9.888054746 -9.891012062 -9.958057736 -9.934023475 -9.780318634
## [17,] 0.891925828 0.667416689 0.824074306 0.742758944 0.760100594
## [18,] -16.982335654 -17.063658187 -17.124050400 -17.109110742 -16.848822668
## [19,] -17.833046655 -18.210923749 -18.074960244 -18.153569344 -17.789153516
## [20,] -17.114113509 -16.639069601 -17.072677911 -16.882265425 -16.740304119
## [21,] -17.551289634 -17.974806134 -17.808380128 -17.897269656 -17.535107377
## [22,] -17.696768526 -17.273530034 -17.606715267 -17.571817639 -17.192614237
## [23,] -17.606986108 -18.320225314 -18.055015421 -18.079122645 -17.903809722
## [24,] -17.287769615 -18.621093489 -16.337528751 -18.463671759 -16.930511477
## [25,] -17.749632990 -16.440267714 -19.165673209 -16.643956626 -18.183921122
## [26,] -17.358853289 -18.540089693 -16.624406000 -18.492542984 -15.210581901
## [27,] -10.561766160 -10.043919600 -11.320556976 -10.258292440 -10.974857848
## [28,] -34.746719211 -35.170410512 -34.725789575 -35.038393514 -35.220983537
## [29,] -5.495845657 -6.404657008 -4.540004353 -5.886168091 -6.465867251
## [30,] -0.250863774 0.083526396 -0.089328554 -0.013691932 -0.172094830
## [31,] -6.874942926 -8.908385584 -7.612960427 -8.261909811 -7.767534510
## [32,] -2.945786647 -1.502361698 -2.412521391 -2.001258250 -2.280792874
## [33,] -5.758430700 -8.248085130 -6.698518738 -7.462336834 -6.981206389
## [34,] -5.726199893 -4.296205297 -5.189839283 -4.796122862 -4.915678870
## [35,] -4.200279876 -7.356390073 -4.342594520 -6.194156578 -5.253330717
## [36,] -5.523287555 -3.934970218 -7.459714937 -4.209337933 -5.341304760
## [37,] -4.584732968 -7.309351758 -3.424850376 -7.994425910 -4.887011101
## [38,] -4.007984739 -4.595929968 -4.189884167 -3.469931997 -9.068748510
## [39,] -1.760146190 -1.876732407 -1.892018531 -2.498395386 0.749059955
## [40,] -12.984627017 -15.179120211 -14.213646146 -13.612986974 -16.317072493
## [41,] 2.478268090 3.437797033 2.236620921 5.149361088 -2.870237163
## [42,] 7.730391474 7.508970037 7.697038372 7.611090688 7.550164276
## [43,] 10.703982481 10.866042958 10.837506854 10.858060475 10.656566556
## [44,] 8.993992747 8.514541711 8.884890270 8.722012101 8.703365255
## [45,] 10.273448475 10.529730931 10.444506913 10.479053349 10.273137925
## [46,] 9.975479491 9.462672273 9.745682867 9.771058045 9.439527686
## [47,] 10.047277005 10.451919076 10.507105593 10.257592879 10.410032593
## [48,] 9.586288837 11.542695841 8.052693230 11.304078581 9.200818213
## [49,] 10.043608641 6.916962059 12.540589659 6.966767639 11.459951263
## [50,] 10.382240371 11.883464118 9.445969320 11.899485740 5.276705317
## [51,] 4.640521011 3.583008387 5.567259481 4.000819995 6.731023521
## [52,] 19.785555383 20.019620271 18.976200988 18.445126711 25.637822411
## [53,] -4.273745388 -3.022066490 -6.516671679 -6.189028939 4.677256558
## [54,] -1.483784665 -1.573599304 -1.537114888 -1.551956238 -1.488946192
## [55,] -0.682559600 -0.116306016 -0.492904747 -0.306699256 -0.401046006
## [56,] -1.149441519 -1.643200222 -1.343596064 -1.473662296 -1.384448510
## [57,] -0.880942795 -0.169387820 -0.625413849 -0.409998589 -0.485679736
## [58,] -0.210755559 -0.718425375 -0.454553784 -0.508029383 -0.575753863
## [59,] -1.551482679 -0.633783644 -1.543308009 -1.002787373 -1.081416446
## [60,] -0.307234976 -1.650388296 0.969868665 -1.303141834 -1.012607588
## [61,] -1.898448023 0.203687910 -3.523129227 0.006643482 -0.037201541
## [62,] -1.184068294 -1.635682417 -0.697188393 -1.823848584 1.789023374
## [63,] -0.402094188 0.066440677 -0.812262602 -0.030083392 -1.613234053
## [64,] -0.815717359 -0.249792986 0.010966215 0.185375875 -3.184982183
## [65,] 2.861113200 1.998142064 4.057942074 3.300665820 -1.115185762
## [66,] -0.007456047 -0.008228841 -0.008471336 -0.008762322 -0.004737211
## [,11] [,12] [,13] [,14] [,15]
## [1,] -7.314751552 -30.182786102 -3.976483671 -17.61639790 -5.45103016
## [2,] 6.219533640 23.004292381 3.790625345 17.05902187 1.27302487
## [3,] 7.867038528 33.940825522 4.044232772 17.69271231 7.68713039
## [4,] 6.853745311 27.281567678 3.903597866 16.90036649 4.20510748
## [5,] 7.608102257 31.888884233 4.011779850 18.07881655 6.15738931
## [6,] 7.137587705 29.282913514 4.026542496 17.50562246 4.95587021
## [7,] 7.516957025 30.653990786 3.735043649 17.64237536 5.78211556
## [8,] 7.117039760 30.071761103 4.318836136 17.67269878 5.25745861
## [9,] 7.691306627 30.131491829 3.326129647 17.67277476 5.53168058
## [10,] 6.371284356 30.262614703 5.374069739 17.37933688 5.35668165
## [11,] 5.122348579 14.397227060 -0.100430496 10.56028996 1.70241370
## [12,] 14.397227060 67.956607566 11.316372592 34.81098856 14.68941622
## [13,] -0.100430496 11.316372592 9.032164607 7.11619584 -0.44917553
## [14,] 10.560289957 34.810988562 7.116195839 64.88544183 -27.78832547
## [15,] 1.702413705 14.689416219 -0.449175533 -27.78832547 37.96342213
## [16,] -6.105559245 -19.176780490 -4.221092531 -38.39096124 16.25089317
## [17,] 0.873219523 0.440160170 1.093447829 17.40002246 -17.83892981
## [18,] -10.238844393 -33.686023935 -6.922325178 -59.79608648 25.85650706
## [19,] -10.732282614 -35.948239903 -7.182802700 -66.88872183 28.59772455
## [20,] -10.308076618 -32.920240386 -7.058127428 -63.36275111 27.73185487
## [21,] -10.606232687 -35.358215506 -7.144674066 -65.33978619 27.83233627
## [22,] -10.602503529 -34.444063600 -7.064493247 -64.81823246 27.93836403
## [23,] -10.620618660 -35.935654272 -7.282198717 -65.26772966 27.43326927
## [24,] -10.750231021 -34.741343508 -6.858484681 -64.72683741 27.67222708
## [25,] -10.177633755 -34.860281103 -7.651370085 -65.03533956 27.96547269
## [26,] -11.779562041 -34.443070660 -5.359203888 -64.52590106 27.39241645
## [27,] -5.082298417 -21.885361355 -6.257686386 -35.91491407 14.36018360
## [28,] -21.160602082 -68.419401800 -14.450204818 -136.18233156 60.97030035
## [29,] -4.140886857 -10.590480771 -0.670195021 -17.86336368 6.79708489
## [30,] -0.071550570 -0.008893207 0.070960801 25.63857574 -24.56884549
## [31,] -2.342908735 -22.155222848 0.809296952 30.68840031 -44.71399218
## [32,] -0.539880930 -6.089879560 0.514005135 29.86313915 -34.27968827
## [33,] -2.398100042 -19.147467973 0.365844623 26.38778197 -39.61549997
## [34,] -1.523092392 -13.267353479 0.343246230 27.57431010 -36.89427073
## [35,] -1.848160897 -15.009483861 0.793646298 28.20890649 -38.55736032
## [36,] -1.546268600 -14.541005208 -0.068956640 27.30478621 -37.40114172
## [37,] -2.205410910 -14.723181057 1.249348055 27.76702953 -38.08132452
## [38,] 1.007309588 -12.184865848 -0.699667119 30.19196812 -38.27289129
## [39,] -3.405769823 -2.817184067 4.043473049 13.11874520 -16.71592656
## [40,] 0.408142712 -47.881541695 -13.308692175 62.21155248 -88.83755021
## [41,] 10.066203002 -6.346666208 -28.992905909 12.54367224 -7.00023096
## [42,] 5.015075599 13.919880756 3.774830840 31.87525723 -16.09256761
## [43,] 6.541578026 21.237982139 4.420183182 41.09383245 -16.67873866
## [44,] 5.654001680 16.251084199 4.118671649 36.76761580 -16.83483046
## [45,] 6.306757367 20.499197510 4.275797517 39.08887058 -15.97051010
## [46,] 6.098167891 18.596709316 4.198687096 38.36495846 -16.60500867
## [47,] 6.185099766 20.309885403 4.305996306 38.69054035 -15.77919754
## [48,] 6.352810320 19.504522673 4.122648776 38.38894701 -16.08901572
## [49,] 5.235514018 17.761316873 4.274401108 37.44696957 -16.25256101
## [50,] 8.439999234 21.575719262 3.838927752 39.83309290 -16.16121305
## [51,] 1.402010383 8.181139984 1.915534073 15.62916296 -5.47269520
## [52,] 7.406291868 44.272948791 25.164608044 86.56750299 -42.06740700
## [53,] -10.007632607 3.109082594 26.257889124 -9.95251249 1.87053873
## [54,] -0.896359329 -3.088446802 -0.687236204 -14.34257481 12.49679739
## [55,] -0.862980669 0.933863337 -1.285459660 -18.96774971 20.02330368
## [56,] -1.073527961 -2.239071656 -1.067698345 -17.67153189 16.90781355
## [57,] -0.773773415 0.323022899 -1.097372415 -17.25735414 18.20932520
## [58,] -0.715112631 0.128658989 -1.011493769 -16.71217776 17.62995144
## [59,] -1.061617513 -1.392523406 -1.240276869 -18.16193759 17.87662432
## [60,] -0.650712135 0.057070805 -0.885423345 -16.54834486 17.45573754
## [61,] -1.502510999 -1.986500276 -1.429042626 -19.16642493 18.29908490
## [62,] -2.263676744 -1.819440907 -0.910891008 -18.21573252 17.81321303
## [63,] 0.781671838 -0.614716155 -0.714374360 -6.14711475 6.40145985
## [64,] -0.107490091 1.583452527 -7.538866852 -41.45782536 45.30092801
## [65,] 3.957504439 2.109502805 -6.262101788 6.24466357 -0.11857121
## [66,] -0.004955396 -0.018329263 0.004656771 0.01777169 -0.03141706
## [,16] [,17] [,18] [,19] [,20]
## [1,] 9.90965036 -0.75904460 17.06325081 18.06379622 16.90109357
## [2,] -9.84864525 2.00896111 -16.64376268 -17.16686696 -16.93891498
## [3,] -9.77248337 0.11194327 -17.06211592 -18.43731827 -16.45365961
## [4,] -9.63594192 1.01347188 -16.48037216 -16.89624491 -17.28945803
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## [8,] -9.95805774 0.82407431 -17.12405040 -18.07496024 -17.07267791
## [9,] -9.93402348 0.74275894 -17.10911074 -18.15356934 -16.88226543
## [10,] -9.78031863 0.76010059 -16.84882267 -17.78915352 -16.74030412
## [11,] -6.10555925 0.87321952 -10.23884439 -10.73228261 -10.30807662
## [12,] -19.17678049 0.44016017 -33.68602394 -35.94823990 -32.92024039
## [13,] -4.22109253 1.09344783 -6.92232518 -7.18280270 -7.05812743
## [14,] -38.39096124 17.40002246 -59.79608648 -66.88872183 -63.36275111
## [15,] 16.25089317 -17.83892981 25.85650706 28.59772455 27.73185487
## [16,] 23.20033881 -9.70598228 35.22891159 39.43209498 37.70930839
## [17,] -9.70598228 9.53529323 -16.51593003 -17.98882892 -17.06398574
## [18,] 35.22891159 -16.51593003 56.56350359 60.92875793 58.89333779
## [19,] 39.43209498 -17.98882892 60.92875793 70.60070465 63.57107001
## [20,] 37.70930839 -17.06398574 58.89333779 63.57107001 65.39011088
## [21,] 38.57091482 -17.52461487 60.03144176 68.05666073 62.40389187
## [22,] 38.42234884 -17.40242058 59.88863123 66.21641277 64.50725902
## [23,] 38.52425647 -17.37962502 60.04551009 67.76251396 62.73217060
## [24,] 38.30366281 -17.33790308 59.66942200 66.64096942 63.35574957
## [25,] 38.48531802 -17.47508861 59.93333989 67.04999164 63.51606153
## [26,] 38.17248044 -17.21930979 59.47431879 66.48426472 63.05549674
## [27,] 21.19364136 -9.35159126 33.41905652 36.99681064 34.88603274
## [28,] 80.68782614 -37.28040624 124.69068906 140.61228637 133.14418414
## [29,] 10.60156571 -4.52852237 17.02628731 17.80070667 18.29561067
## [30,] -14.40156338 12.69822820 -23.03513774 -27.43960684 -23.62039366
## [31,] -18.72662620 20.05355081 -28.97442277 -29.07335596 -34.36158143
## [32,] -16.77671972 17.23671739 -27.28601735 -33.68677364 -23.70017637
## [33,] -15.71657347 18.05997765 -24.73914249 -26.11629309 -28.63150415
## [34,] -15.98744437 17.52045679 -25.48086343 -29.18134488 -25.86729629
## [35,] -16.57080110 18.05638711 -26.37487125 -28.48726048 -29.22860803
## [36,] -15.90572823 17.62405756 -25.34003212 -28.41329891 -26.60349363
## [37,] -16.28379172 17.84223764 -25.89558238 -28.31897840 -28.22964345
## [38,] -17.57866466 18.29073143 -28.00409936 -31.45926792 -29.31316417
## [39,] -7.19174036 8.32094039 -12.03786027 -13.64775736 -12.84879106
## [40,] -37.63523575 40.46558578 -58.37672059 -63.26255210 -63.45843638
## [41,] -6.62406177 4.65921773 -11.32771499 -13.45880768 -11.30368321
## [42,] -19.22635114 9.30263171 -30.16595715 -32.39333937 -31.74503306
## [43,] -24.79029628 10.09741696 -37.38269946 -42.67804153 -39.69298918
## [44,] -22.45226409 9.59491901 -33.98407998 -36.67364926 -38.43582481
## [45,] -23.51869376 9.74769463 -35.78072845 -40.58390151 -37.45761386
## [46,] -23.24220447 9.79623176 -35.29346920 -39.02507154 -38.48246504
## [47,] -23.29841720 9.62394471 -35.44405526 -40.07107984 -37.27974635
## [48,] -23.17307342 9.66966434 -35.20351638 -39.53275706 -37.48001203
## [49,] -22.71517458 9.54053463 -34.43156148 -38.12755587 -37.47892220
## [50,] -23.96717899 9.90160416 -36.50779930 -41.20960870 -38.51074098
## [51,] -9.56035523 3.37051174 -14.43444105 -15.93542142 -15.48966654
## [52,] -52.31899819 24.04020200 -79.32690609 -88.38591217 -86.37827752
## [53,] 5.45055081 -2.19853094 8.86454753 11.54854651 7.26872745
## [54,] 7.83051162 -7.05505959 13.23792389 15.14105989 13.52670452
## [55,] 10.80696903 -10.40701698 18.13116721 18.92320997 19.67238041
## [56,] 9.68979579 -9.34291949 16.58203747 19.21750231 15.40656316
## [57,] 9.68407299 -9.61126523 16.45239638 17.52037325 17.59970623
## [58,] 9.30385038 -9.37014650 15.80886995 17.51511428 15.95035160
## [59,] 10.12770695 -9.66296320 17.26686240 18.66062294 17.99300722
## [60,] 9.19086733 -9.30962349 15.70608710 17.19709034 16.04424628
## [61,] 10.73075572 -9.91189735 18.14378202 19.86527842 18.69339149
## [62,] 10.13813401 -9.65634933 17.23654066 19.00826577 17.49403634
## [63,] 3.19208780 -3.60310090 5.78056256 6.45054139 5.87038487
## [64,] 23.69848309 -23.38242511 39.64783805 42.20673102 41.93658395
## [65,] -3.96736832 0.43674108 -5.68389550 -6.89088737 -5.13773343
## [66,] -0.01086202 0.01357292 -0.01648165 -0.01747131 -0.01945263
## [,21] [,22] [,23] [,24] [,25]
## [1,] 17.80411923 17.51488480 18.02455142 17.6121287 17.5815595
## [2,] -17.04792993 -17.12466414 -17.13534270 -17.0484862 -17.0674775
## [3,] -18.06328437 -17.42418007 -18.36948310 -17.6904543 -17.6246432
## [4,] -16.69113870 -17.23569454 -16.83407280 -16.8935880 -16.9273335
## [5,] -18.41999715 -17.70447734 -18.81974099 -18.1181473 -17.9617856
## [6,] -17.55128963 -17.69676853 -17.60698611 -17.2877696 -17.7496330
## [7,] -17.97480613 -17.27353003 -18.32022531 -18.6210935 -16.4402677
## [8,] -17.80838013 -17.60671527 -18.05501542 -16.3375288 -19.1656732
## [9,] -17.89726966 -17.57181764 -18.07912265 -18.4636718 -16.6439566
## [10,] -17.53510738 -17.19261424 -17.90380972 -16.9305115 -18.1839211
## [11,] -10.60623269 -10.60250353 -10.62061866 -10.7502310 -10.1776338
## [12,] -35.35821551 -34.44406360 -35.93565427 -34.7413435 -34.8602811
## [13,] -7.14467407 -7.06449325 -7.28219872 -6.8584847 -7.6513701
## [14,] -65.33978619 -64.81823246 -65.26772966 -64.7268374 -65.0353396
## [15,] 27.83233627 27.93836403 27.43326927 27.6722271 27.9654727
## [16,] 38.57091482 38.42234884 38.52425647 38.3036628 38.4853180
## [17,] -17.52461487 -17.40242058 -17.37962502 -17.3379031 -17.4750886
## [18,] 60.03144176 59.88863123 60.04551009 59.6694220 59.9333399
## [19,] 68.05666073 66.21641277 67.76251396 66.6409694 67.0499916
## [20,] 62.40389187 64.50725902 62.73217060 63.3557496 63.5160615
## [21,] 66.79455683 64.34847261 66.39410581 65.0601877 65.4977471
## [22,] 64.34847261 66.98650129 63.56590053 65.1372554 64.6329766
## [23,] 66.39410581 63.56590053 67.66258424 64.5568672 65.7334364
## [24,] 65.06018772 65.13725543 64.55686717 68.2745818 60.3942434
## [25,] 65.49774712 64.63297665 65.73343637 60.3942434 71.8257773
## [26,] 64.92840096 64.88497041 64.43197806 67.6914974 59.0499666
## [27,] 36.30626939 35.69327073 36.37066193 33.8827948 39.0805300
## [28,] 137.00239478 135.97480549 136.97967261 136.2474667 136.1238094
## [29,] 17.49042339 18.00657771 17.84776410 19.9005857 15.0677905
## [30,] -26.42807738 -25.18554835 -26.35002796 -25.4936224 -25.7217554
## [31,] -29.19366724 -32.18140545 -28.81437604 -30.7256543 -30.9157709
## [32,] -32.46084384 -27.37637613 -31.98136158 -29.5203935 -29.9345888
## [33,] -25.35848072 -27.99883241 -24.53924612 -26.3651706 -26.6556490
## [34,] -28.50451345 -25.67681488 -28.91074208 -27.3965044 -27.6077532
## [35,] -27.63439808 -29.80297265 -26.90714667 -26.9812179 -29.8270358
## [36,] -27.72327689 -26.85884213 -27.59951717 -30.9184762 -22.8896512
## [37,] -27.48905549 -28.19716819 -26.99190353 -23.1852037 -33.0966689
## [38,] -30.69082913 -30.60403443 -29.49640137 -31.2746904 -27.7870174
## [39,] -13.18688225 -12.90276413 -13.45792765 -12.8948131 -13.7436049
## [40,] -61.74005619 -63.12235510 -60.41379247 -61.8673445 -62.4748166
## [41,] -12.98610753 -12.64981251 -12.30779497 -12.3998668 -11.8572360
## [42,] -31.86951647 -32.01429820 -31.77494475 -31.8104871 -31.9687667
## [43,] -41.54632962 -40.89642288 -41.46049716 -40.9695178 -41.1968910
## [44,] -36.02529432 -37.54946305 -36.11209149 -36.8031784 -36.8374931
## [45,] -39.78908033 -38.60097959 -39.72047262 -38.8978243 -39.2337270
## [46,] -38.02150693 -40.15574716 -37.12483046 -38.7998351 -37.9519289
## [47,] -39.27716597 -37.39736016 -40.28778121 -37.7967610 -39.6311828
## [48,] -38.70844746 -38.61173517 -38.46148227 -43.7131281 -31.4861826
## [49,] -37.16614914 -37.29451418 -37.40850313 -28.0071489 -50.6646979
## [50,] -40.38023453 -40.50147880 -39.39120254 -45.2156620 -30.4388529
## [51,] -15.61994659 -15.30575203 -15.94175096 -11.7078100 -22.0239356
## [52,] -86.19497617 -86.39583259 -87.21111447 -87.4410801 -86.4858550
## [53,] 11.39659734 10.04695025 9.58893104 5.3456297 14.9755756
## [54,] 14.65393054 14.17588900 14.59849016 14.2776996 14.3872299
## [55,] 18.65829305 19.38298623 18.52605198 18.9499630 19.0550485
## [56,] 18.61100964 16.72761365 18.41562686 17.5286206 17.7257520
## [57,] 17.04448440 17.80598061 16.77641907 17.2459332 17.3329748
## [58,] 17.12445499 15.54270313 17.49941233 16.4507335 16.9525231
## [59,] 18.12732969 18.94339213 17.77344531 17.7845087 18.6374605
## [60,] 16.78650914 16.26907692 16.89212420 19.6582708 12.6994936
## [61,] 19.33285385 19.33704200 18.76034155 14.3113372 25.0088665
## [62,] 18.56112659 18.51877724 17.91389070 20.6801375 13.8692889
## [63,] 6.24342467 5.92463332 6.37432723 4.3990471 8.9800910
## [64,] 41.10686740 41.53758042 41.26232123 41.7942708 41.5625989
## [65,] -6.90194958 -6.36586266 -6.01770489 -4.2244820 -8.4007981
## [66,] -0.01696038 -0.01812897 -0.01724008 -0.0182767 -0.0177667
## [,26] [,27] [,28] [,29] [,30]
## [1,] 17.66253304 1.055334e+01 34.93378004 5.63952633 0.063331178
## [2,] -17.03626084 -9.978990e+00 -34.39438914 -5.49972354 -0.849283800
## [3,] -17.78288156 -1.074471e+01 -34.75630821 -5.57407574 0.365383473
## [4,] -16.91116207 -1.002805e+01 -33.70451178 -5.58527849 -0.994651534
## [5,] -18.17870030 -1.086486e+01 -35.74600956 -5.74276977 0.438597800
## [6,] -17.35885329 -1.056177e+01 -34.74671921 -5.49584566 -0.250863774
## [7,] -18.54008969 -1.004392e+01 -35.17041051 -6.40465701 0.083526396
## [8,] -16.62440600 -1.132056e+01 -34.72578958 -4.54000435 -0.089328554
## [9,] -18.49254298 -1.025829e+01 -35.03839351 -5.88616809 -0.013691932
## [10,] -15.21058190 -1.097486e+01 -35.22098354 -6.46586725 -0.172094830
## [11,] -11.77956204 -5.082298e+00 -21.16060208 -4.14088686 -0.071550570
## [12,] -34.44307066 -2.188536e+01 -68.41940180 -10.59048077 -0.008893206
## [13,] -5.35920389 -6.257686e+00 -14.45020482 -0.67019502 0.070960801
## [14,] -64.52590106 -3.591491e+01 -136.18233156 -17.86336368 25.638575740
## [15,] 27.39241645 1.436018e+01 60.97030035 6.79708489 -24.568845492
## [16,] 38.17248044 2.119364e+01 80.68782614 10.60156571 -14.401563377
## [17,] -17.21930979 -9.351591e+00 -37.28040624 -4.52852237 12.698228205
## [18,] 59.47431879 3.341906e+01 124.69068906 17.02628731 -23.035137742
## [19,] 66.48426472 3.699681e+01 140.61228637 17.80070667 -27.439606845
## [20,] 63.05549674 3.488603e+01 133.14418414 18.29561067 -23.620393663
## [21,] 64.92840096 3.630627e+01 137.00239478 17.49042339 -26.428077380
## [22,] 64.88497041 3.569327e+01 135.97480549 18.00657771 -25.185548355
## [23,] 64.43197806 3.637066e+01 136.97967261 17.84776410 -26.350027959
## [24,] 67.69149736 3.388279e+01 136.24746674 19.90058570 -25.493622380
## [25,] 59.04996656 3.908053e+01 136.12380935 15.06779054 -25.721755353
## [26,] 74.73252033 2.963171e+01 136.05335595 23.19579908 -25.430833373
## [27,] 29.63171154 3.068008e+01 70.02782238 -5.65194317 -13.942064624
## [28,] 136.05335595 7.002782e+01 294.29358538 52.76401097 -54.722823591
## [29,] 23.19579908 -5.651943e+00 52.76401097 43.44433059 -5.886117260
## [30,] -25.43083337 -1.394206e+01 -54.72282359 -5.88611726 27.453213273
## [31,] -30.22543477 -1.533554e+01 -68.19705814 -8.64766886 22.513564450
## [32,] -29.42302209 -1.617493e+01 -64.22671273 -6.13491643 29.473021532
## [33,] -26.00254591 -1.322538e+01 -58.63200095 -7.09500032 22.403513662
## [34,] -27.09897707 -1.438787e+01 -60.35979206 -6.42459970 25.074546494
## [35,] -26.96222031 -1.514491e+01 -61.63428334 -6.21470579 24.455956681
## [36,] -29.36955246 -1.139080e+01 -62.05011642 -12.15445987 24.645420379
## [37,] -25.85391045 -1.592377e+01 -59.98776414 -4.11376722 24.302206694
## [38,] -37.14352122 -1.689959e+01 -57.54710678 7.63955531 26.039881097
## [39,] -10.03123080 -9.681124e+00 -31.54821910 -6.43055613 14.072872007
## [40,] -62.27703438 -3.172795e+01 -133.54244187 -8.93708545 48.695090966
## [41,] -16.32166860 -5.137283e+00 -17.39311684 10.39292738 11.220728776
## [42,] -31.65893308 -1.760383e+01 -66.94985073 -9.09126373 10.426735768
## [43,] -40.85472958 -2.271043e+01 -86.36651385 -11.14588328 16.177077226
## [44,] -36.57611685 -2.005046e+01 -77.63783462 -10.76320513 12.314223549
## [45,] -38.81802860 -2.172047e+01 -81.97262108 -10.49950410 15.312352693
## [46,] -38.68311871 -2.107174e+01 -80.40676313 -10.27975858 14.071796134
## [47,] -37.66296030 -2.158466e+01 -81.55065371 -11.06409398 14.983095944
## [48,] -43.14877445 -1.865876e+01 -80.75712676 -12.53087652 14.508929974
## [49,] -27.04863607 -2.355053e+01 -81.84942632 -13.62023038 13.650089160
## [50,] -58.53663642 -1.614624e+01 -77.41223776 -4.56820256 15.403424786
## [51,] -2.61444631 -2.183355e+01 -30.27823495 8.80459724 4.939051255
## [52,] -84.62687826 -3.262130e+01 -211.50780394 -82.88175107 33.961548770
## [53,] 3.19968903 4.008214e+01 -29.83532631 -107.78781814 -5.695130849
## [54,] 14.22700876 7.842250e+00 30.45089517 3.57484164 -12.189586982
## [55,] 18.77600892 1.006777e+01 40.83407243 5.22571818 -12.575682727
## [56,] 17.43639154 9.638781e+00 37.66617487 4.34562349 -13.872805205
## [57,] 17.12332127 9.208407e+00 37.05119434 4.59322078 -12.299795490
## [58,] 16.26665129 8.932469e+00 36.10231331 4.64952499 -12.296899699
## [59,] 17.89450987 1.020898e+01 38.29241269 3.63475523 -13.243580031
## [60,] 18.15473300 5.722280e+00 38.68322521 11.91590690 -12.385852171
## [61,] 17.67400566 1.604992e+01 33.97534836 -10.84621480 -13.529636211
## [62,] 27.31402948 6.886313e+00 35.66709581 1.24035376 -13.288425822
## [63,] 0.05439311 1.002946e+01 11.61373668 -5.66482367 -5.841817358
## [64,] 40.19993084 1.456476e+01 103.36416120 40.08575353 -27.583993866
## [65,] -3.30580031 -2.096014e+01 11.98167297 53.00660566 0.519300998
## [66,] -0.01673708 5.567313e-04 -0.05578447 -0.03749309 0.016895861
## [,31] [,32] [,33] [,34] [,35]
## [1,] 8.04031212 2.16836939 7.2092625 4.93601916 5.56844828
## [2,] -1.01604078 -0.64724824 -1.7687957 -1.28026914 -1.16390847
## [3,] -12.48538643 -2.29934532 -10.4156081 -6.71212547 -8.02975996
## [4,] -4.59417937 -4.35522141 -4.4298951 -4.76131559 -3.68058346
## [5,] -9.81030770 -1.27451979 -9.1238741 -4.73702314 -6.81088244
## [6,] -6.87494293 -2.94578665 -5.7584307 -5.72619989 -4.20027988
## [7,] -8.90838558 -1.50236170 -8.2480851 -4.29620530 -7.35639007
## [8,] -7.61296043 -2.41252139 -6.6985187 -5.18983928 -4.34259452
## [9,] -8.26190981 -2.00125825 -7.4623368 -4.79612286 -6.19415658
## [10,] -7.76753451 -2.28079287 -6.9812064 -4.91567887 -5.25333072
## [11,] -2.34290873 -0.53988093 -2.3981000 -1.52309239 -1.84816090
## [12,] -22.15522285 -6.08987956 -19.1474680 -13.26735348 -15.00948386
## [13,] 0.80929695 0.51400513 0.3658446 0.34324623 0.79364630
## [14,] 30.68840031 29.86313916 26.3877820 27.57431010 28.20890649
## [15,] -44.71399218 -34.27968827 -39.6155000 -36.89427073 -38.55736032
## [16,] -18.72662620 -16.77671972 -15.7165735 -15.98744437 -16.57080110
## [17,] 20.05355081 17.23671739 18.0599777 17.52045679 18.05638711
## [18,] -28.97442277 -27.28601735 -24.7391425 -25.48086343 -26.37487125
## [19,] -29.07335596 -33.68677365 -26.1162931 -29.18134488 -28.48726048
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## [30,] 22.51356445 29.47302153 22.4035137 25.07454649 24.45595668
## [31,] 62.22107599 29.66445400 50.8453271 41.16339419 46.72557556
## [32,] 29.66445400 48.86686702 28.9628171 38.56323395 31.45508738
## [33,] 50.84532715 28.96281714 45.8801357 34.80359610 42.56894397
## [34,] 41.16339419 38.56323395 34.8035961 41.31332958 33.69889116
## [35,] 46.72557556 31.45508738 42.5689440 33.69889116 43.37698371
## [36,] 43.16759291 35.70374333 37.7078472 38.29466273 34.52938103
## [37,] 45.53204723 32.98784642 40.6479623 35.74898481 41.39111549
## [38,] 43.63917824 36.29723133 38.9324347 37.61730540 38.55119135
## [39,] 18.18711399 16.57251567 16.9170587 16.37705242 16.96442703
## [40,] 109.58838954 75.03900338 94.8952380 85.54139207 90.33110374
## [41,] 4.42503511 10.71133486 5.1041701 7.76511348 6.26698516
## [42,] 19.69460833 14.90283801 16.3966024 15.59860681 16.46873179
## [43,] 18.31722424 18.73948848 15.5324897 16.75679216 16.80296139
## [44,] 22.81278197 12.11134845 18.3361407 15.54326456 17.72774551
## [45,] 16.93856392 18.66239247 14.3782362 16.30094149 15.94952089
## [46,] 20.17634486 14.94351702 17.4442010 14.40771898 18.20874222
## [47,] 16.95389898 18.42605277 13.8518508 17.11238281 15.34408508
## [48,] 18.21372193 17.24770531 15.1436402 16.34650071 14.25828275
## [49,] 19.75771945 15.14599700 16.6890331 15.04250870 19.98773499
## [50,] 17.45096718 17.95746315 14.9222181 15.99375190 15.33254660
## [51,] 6.57647661 5.60655590 5.2767453 5.46712375 6.44172110
## [52,] 50.99988861 39.84478947 42.7925393 40.67866599 42.53284488
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## [65,] -1.04632674 1.90842344 -1.0647720 0.54493718 0.87669277
## [66,] 0.04059909 0.02396129 0.0350248 0.02948793 0.03205771
## [,36] [,37] [,38] [,39] [,40]
## [1,] 5.36380120 5.54788500 4.369824270 1.87025657 14.41117105
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## [3,] -7.41545691 -7.90240172 -6.329346949 -2.37628965 -21.00923451
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## [11,] -1.54626860 -2.20541091 1.007309588 -3.40576982 0.40814271
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## [13,] -0.06895664 1.24934806 -0.699667118 4.04347305 -13.30869217
## [14,] 27.30478621 27.76702953 30.191968121 13.11874520 62.21155248
## [15,] -37.40114172 -38.08132452 -38.272891290 -16.71592655 -88.83755021
## [16,] -15.90572823 -16.28379172 -17.578664664 -7.19174036 -37.63523575
## [17,] 17.62405756 17.84223764 18.290731429 8.32094039 40.46558578
## [18,] -25.34003212 -25.89558238 -28.004099362 -12.03786027 -58.37672059
## [19,] -28.41329891 -28.31897840 -31.459267925 -13.64775736 -63.26255210
## [20,] -26.60349363 -28.22964345 -29.313164168 -12.84879106 -63.45843639
## [21,] -27.72327689 -27.48905549 -30.690829132 -13.18688225 -61.74005619
## [22,] -26.85884213 -28.19716819 -30.604034432 -12.90276413 -63.12235511
## [23,] -27.59951717 -26.99190353 -29.496401370 -13.45792765 -60.41379247
## [24,] -30.91847622 -23.18520368 -31.274690399 -12.89481309 -61.86734450
## [25,] -22.88965122 -33.09666888 -27.787017441 -13.74360489 -62.47481664
## [26,] -29.36955246 -25.85391045 -37.143521217 -10.03123080 -62.27703438
## [27,] -11.39079823 -15.92377088 -16.899585057 -9.68112412 -31.72794841
## [28,] -62.05011643 -59.98776414 -57.547106777 -31.54821910 -133.54244187
## [29,] -12.15445987 -4.11376722 7.639555310 -6.43055613 -8.93708545
## [30,] 24.64542038 24.30220669 26.039881097 14.07287201 48.69509097
## [31,] 43.16759290 45.53204723 43.639178237 18.18711399 109.58838955
## [32,] 35.70374333 32.98784642 36.297231327 16.57251567 75.03900338
## [33,] 37.70784715 40.64796232 38.932434723 16.91705875 94.89523800
## [34,] 38.29466273 35.74898481 37.617305402 16.37705242 85.54139207
## [35,] 34.52938103 41.39111549 38.551191346 16.96442703 90.33110374
## [36,] 44.74133891 30.24907466 34.969371452 17.95064231 86.88805987
## [37,] 30.24907466 53.50579137 37.106992212 17.95998186 89.03300877
## [38,] 34.96937145 37.10699221 72.766719538 0.11985375 89.63525390
## [39,] 17.95064231 17.95998186 0.119853751 22.36091226 28.44287349
## [40,] 86.88805987 89.03300877 89.635253901 28.44287349 258.26810285
## [41,] 5.68339819 5.35516666 21.308417726 -14.70624261 81.45070999
## [42,] 15.69585456 16.18486000 16.952886150 6.27861082 39.53811111
## [43,] 16.44481625 16.61850647 18.266928336 7.67500908 37.74575087
## [44,] 16.10250276 17.16180166 17.436272615 7.01844946 40.68355037
## [45,] 15.82783341 15.86800682 17.613146512 7.26229450 36.22350325
## [46,] 15.92736152 16.54501828 18.753158751 6.86815205 38.97290008
## [47,] 15.43639195 16.27170355 15.969387736 7.68767850 35.76078267
## [48,] 22.09384828 8.27637480 20.378849209 6.63817077 36.48465159
## [49,] 6.47259201 31.17644252 3.912411488 11.50795186 38.50752483
## [50,] 16.74504392 14.85593113 51.180194070 -5.99738821 33.99802319
## [51,] 2.78951926 7.02329381 -8.513482253 10.83951305 20.21631093
## [52,] 49.44702015 39.83462824 3.455571099 42.71533085 45.51935600
## [53,] 13.91019522 -6.84945698 -62.819571919 33.82107144 -96.99848941
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## [58,] -18.07759833 -17.35313936 -17.501128908 -8.35768529 -40.08837606
## [59,] -16.38584903 -18.79173262 -19.359672443 -8.06461748 -40.29159417
## [60,] -23.19907478 -11.84251590 -12.044813235 -10.94642423 -38.43100510
## [61,] -9.35243559 -29.78126121 -32.613475469 -2.66022501 -43.73505015
## [62,] -17.68918632 -16.89772694 -38.097966812 0.17723508 -37.81076238
## [63,] -5.24766850 -7.36950697 1.398106574 -9.14681797 -15.97420039
## [64,] -48.89222936 -44.17273963 -23.409117400 -31.06563512 -90.26163628
## [65,] -8.06812038 2.99766431 33.165291489 -16.86337213 33.67233577
## [66,] 0.03553578 0.03057162 0.006151924 0.02548176 0.05683107
## [,41] [,42] [,43] [,44] [,45]
## [1,] -2.86658888 -7.61580769 -10.81463740 -8.75305173 -10.432200160
## [2,] 1.49501734 8.62992481 10.35190179 9.51032175 10.004887148
## [3,] 3.62797769 6.98914328 10.87800005 7.98796127 10.573272915
## [4,] 1.98885872 7.82260262 10.22989458 9.48332304 9.739540569
## [5,] 3.40716616 7.53563556 11.15431317 8.49726483 10.862393981
## [6,] 2.47826809 7.73039147 10.70398248 8.99399275 10.273448476
## [7,] 3.43779703 7.50897004 10.86604296 8.51454171 10.529730931
## [8,] 2.23662092 7.69703837 10.83750685 8.88489027 10.444506913
## [9,] 5.14936109 7.61109069 10.85806047 8.72201210 10.479053350
## [10,] -2.87023716 7.55016428 10.65656656 8.70336526 10.273137925
## [11,] 10.06620300 5.01507560 6.54157803 5.65400168 6.306757367
## [12,] -6.34666621 13.91988076 21.23798214 16.25108420 20.499197511
## [13,] -28.99290591 3.77483084 4.42018318 4.11867165 4.275797518
## [14,] 12.54367224 31.87525723 41.09383245 36.76761580 39.088870576
## [15,] -7.00023096 -16.09256761 -16.67873866 -16.83483046 -15.970510104
## [16,] -6.62406177 -19.22635114 -24.79029628 -22.45226409 -23.518693759
## [17,] 4.65921773 9.30263171 10.09741696 9.59491901 9.747694626
## [18,] -11.32771499 -30.16595715 -37.38269946 -33.98407998 -35.780728453
## [19,] -13.45880768 -32.39333937 -42.67804153 -36.67364926 -40.583901513
## [20,] -11.30368321 -31.74503306 -39.69298918 -38.43582481 -37.457613860
## [21,] -12.98610753 -31.86951647 -41.54632962 -36.02529432 -39.789080329
## [22,] -12.64981251 -32.01429820 -40.89642288 -37.54946305 -38.600979587
## [23,] -12.30779497 -31.77494475 -41.46049716 -36.11209149 -39.720472622
## [24,] -12.39986675 -31.81048709 -40.96951784 -36.80317839 -38.897824276
## [25,] -11.85723602 -31.96876669 -41.19689098 -36.83749310 -39.233727035
## [26,] -16.32166860 -31.65893308 -40.85472958 -36.57611685 -38.818028605
## [27,] -5.13728284 -17.60383116 -22.71043218 -20.05045991 -21.720468719
## [28,] -17.39311684 -66.94985073 -86.36651385 -77.63783462 -81.972621082
## [29,] 10.39292738 -9.09126373 -11.14588328 -10.76320513 -10.499504100
## [30,] 11.22072878 10.42673577 16.17707723 12.31422355 15.312352693
## [31,] 4.42503511 19.69460833 18.31722424 22.81278197 16.938563917
## [32,] 10.71133486 14.90283801 18.73948848 12.11134845 18.662392466
## [33,] 5.10417007 16.39660236 15.53248966 18.33614069 14.378236185
## [34,] 7.76511348 15.59860681 16.75679216 15.54326456 16.300941492
## [35,] 6.26698516 16.46873179 16.80296139 17.72774551 15.949520891
## [36,] 5.68339819 15.69585456 16.44481625 16.10250276 15.827833411
## [37,] 5.35516666 16.18486000 16.61850647 17.16180166 15.868006820
## [38,] 21.30841773 16.95288615 18.26692834 17.43627261 17.613146512
## [39,] -14.70624261 6.27861082 7.67500908 7.01844946 7.262294499
## [40,] 81.45070999 39.53811111 37.74575087 40.68355037 36.223503253
## [41,] 143.88144816 4.34075052 7.58968612 5.19789796 7.237445474
## [42,] 4.34075052 17.52160533 20.03303791 19.17921522 19.239343511
## [43,] 7.58968612 20.03303791 26.78209465 23.44866496 25.351930362
## [44,] 5.19789796 19.17921522 23.44866496 23.67717620 21.943878625
## [45,] 7.23744547 19.23934351 25.35193036 21.94387862 24.264901403
## [46,] 6.79503047 19.39287080 24.65427284 23.06471180 23.157275900
## [47,] 6.45534779 19.09790062 25.08155813 21.90192951 24.031219734
## [48,] 6.04650217 19.15593089 24.80036823 22.30440991 23.513676843
## [49,] 3.62746767 18.96897001 24.14486978 22.43506564 22.862711311
## [50,] 12.01181690 19.63948677 25.73994229 22.65093357 24.489599497
## [51,] 8.74897770 7.91968632 10.19658416 9.32102581 9.697233360
## [52,] -92.95808538 44.01007085 55.56907621 51.98576820 52.411297880
## [53,] -191.34860363 -3.66653496 -6.41801231 -3.00153515 -6.589429539
## [54,] -5.29759247 -6.38806370 -8.54499301 -7.05078736 -8.168622248
## [55,] -4.18627607 -10.64961597 -10.99394295 -11.62791222 -10.455512911
## [56,] -5.47757751 -8.89181529 -10.53149530 -8.07959879 -10.320227041
## [57,] -4.33767046 -9.43204901 -9.91803407 -10.07473107 -9.465458731
## [58,] -4.25542525 -9.02918376 -9.73838728 -9.12566911 -9.439589606
## [59,] -5.37830997 -9.53343815 -10.53977114 -9.89195500 -10.192352765
## [60,] -0.61761152 -8.87338955 -9.57833618 -9.04581838 -9.253911751
## [61,] -13.53495910 -10.06236570 -11.23062224 -10.37371523 -10.893896494
## [62,] -7.34713796 -9.53571556 -10.63760037 -9.68898232 -10.308716213
## [63,] -4.22417863 -2.77410416 -3.44003596 -2.93257698 -3.320370219
## [64,] 29.96119654 -23.94428489 -24.11782718 -24.79954358 -23.178414599
## [65,] 66.39803276 3.24068641 4.36568526 3.03204762 4.411430675
## [66,] -0.03888437 0.01172314 0.01063061 0.01297802 0.009886694
## [,46] [,47] [,48] [,49] [,50]
## [1,] -9.70340843 -10.34454425 -10.01935669 -9.38808607 -10.733366052
## [2,] 9.85384600 9.94594849 9.85860838 9.59458354 10.301237754
## [3,] 9.38249886 10.47058831 9.96507361 9.03957676 10.856786632
## [4,] 9.84115083 9.63137229 9.58313625 9.47903435 10.169844010
## [5,] 9.59354725 10.87864666 10.37308281 9.28979779 11.119702623
## [6,] 9.97547949 10.04727701 9.58628884 10.04360864 10.382240371
## [7,] 9.46267227 10.45191908 11.54269584 6.91696206 11.883464118
## [8,] 9.74568287 10.50710559 8.05269323 12.54058966 9.445969320
## [9,] 9.77105805 10.25759288 11.30407858 6.96676764 11.899485740
## [10,] 9.43952769 10.41003259 9.20081821 11.45995126 5.276705317
## [11,] 6.09816789 6.18509977 6.35281032 5.23551402 8.439999235
## [12,] 18.59670932 20.30988540 19.50452267 17.76131687 21.575719263
## [13,] 4.19868710 4.30599631 4.12264878 4.27440111 3.838927752
## [14,] 38.36495846 38.69054035 38.38894701 37.44696957 39.833092903
## [15,] -16.60500867 -15.77919754 -16.08901572 -16.25256101 -16.161213052
## [16,] -23.24220447 -23.29841720 -23.17307342 -22.71517458 -23.967178992
## [17,] 9.79623176 9.62394471 9.66966434 9.54053463 9.901604157
## [18,] -35.29346920 -35.44405526 -35.20351638 -34.43156148 -36.507799298
## [19,] -39.02507154 -40.07107984 -39.53275706 -38.12755587 -41.209608704
## [20,] -38.48246504 -37.27974635 -37.48001203 -37.47892220 -38.510740982
## [21,] -38.02150693 -39.27716597 -38.70844746 -37.16614914 -40.380234529
## [22,] -40.15574716 -37.39736016 -38.61173517 -37.29451418 -40.501478799
## [23,] -37.12483046 -40.28778121 -38.46148227 -37.40850313 -39.391202535
## [24,] -38.79983506 -37.79676097 -43.71312814 -28.00714886 -45.215662034
## [25,] -37.95192892 -39.63118285 -31.48618257 -50.66469791 -30.438852936
## [26,] -38.68311871 -37.66296030 -43.14877445 -27.04863607 -58.536636423
## [27,] -21.07173568 -21.58466412 -18.65875824 -23.55052525 -16.146238677
## [28,] -80.40676313 -81.55065371 -80.75712676 -81.84942632 -77.412237761
## [29,] -10.27975858 -11.06409398 -12.53087652 -13.62023038 -4.568202561
## [30,] 14.07179613 14.98309594 14.50892997 13.65008916 15.403424786
## [31,] 20.17634486 16.95389898 18.21372193 19.75771945 17.450967177
## [32,] 14.94351702 18.42605277 17.24770531 15.14599700 17.957463154
## [33,] 17.44420097 13.85185085 15.14364022 16.68903311 14.922218067
## [34,] 14.40771898 17.11238281 16.34650071 15.04250870 15.993751904
## [35,] 18.20874222 15.34408508 14.25828275 19.98773499 15.332546597
## [36,] 15.92736152 15.43639195 22.09384828 6.47259201 16.745043924
## [37,] 16.54501827 16.27170355 8.27637480 31.17644252 14.855931131
## [38,] 18.75315875 15.96938774 20.37884921 3.91241149 51.180194069
## [39,] 6.86815205 7.68767850 6.63817077 11.50795186 -5.997388206
## [40,] 38.97290008 35.76078267 36.48465159 38.50752483 33.998023193
## [41,] 6.79503047 6.45534779 6.04650217 3.62746767 12.011816902
## [42,] 19.39287080 19.09790062 19.15593089 18.96897001 19.639486775
## [43,] 24.65427284 25.08155813 24.80036823 24.14486978 25.739942286
## [44,] 23.06471180 21.90192951 22.30440991 22.43506564 22.650933567
## [45,] 23.15727590 24.03121973 23.51367684 22.86271131 24.489599497
## [46,] 25.06426348 21.91432923 23.74149047 21.74602732 25.351803813
## [47,] 21.91432923 24.84949295 22.34534674 24.56288922 22.147424914
## [48,] 23.74149047 22.34534674 32.08636428 6.37108249 33.629957233
## [49,] 21.74602732 24.56288922 6.37108249 55.94526873 -2.167542825
## [50,] 25.35180381 22.14742491 33.62995723 -2.16754282 78.517441876
## [51,] 8.94841258 10.54221500 3.09956167 23.11673675 -20.789948488
## [52,] 51.32794621 53.92797236 51.99504554 66.13532904 22.872384540
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## [65,] 4.83865638 2.88147323 2.66625403 -5.95821008 24.848713834
## [66,] 0.01081028 0.01095106 0.01026279 0.02036393 -0.009209929
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## [3,] 4.416550536 18.589460 -5.3027477 -1.596932821 0.83290584
## [4,] 4.380063461 19.760808 -3.2870604 -1.225519804 -1.19873396
## [5,] 4.516162380 19.727293 -4.9031496 -1.701071509 0.04951674
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## [7,] 3.583008387 20.019620 -3.0220665 -1.573599304 -0.11630602
## [8,] 5.567259481 18.976201 -6.5166717 -1.537114888 -0.49290475
## [9,] 4.000819996 18.445127 -6.1890289 -1.551956239 -0.30669926
## [10,] 6.731023521 25.637822 4.6772566 -1.488946193 -0.40104601
## [11,] 1.402010383 7.406292 -10.0076326 -0.896359329 -0.86298067
## [12,] 8.181139984 44.272949 3.1090826 -3.088446802 0.93386334
## [13,] 1.915534073 25.164608 26.2578891 -0.687236204 -1.28545966
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## [15,] -5.472695204 -42.067407 1.8705387 12.496797386 20.02330368
## [16,] -9.560355227 -52.318998 5.4505508 7.830511621 10.80696903
## [17,] 3.370511739 24.040202 -2.1985309 -7.055059594 -10.40701698
## [18,] -14.434441050 -79.326906 8.8645475 13.237923895 18.13116721
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## [21,] -15.619946591 -86.194976 11.3965973 14.653930545 18.65829305
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## [26,] -2.614446306 -84.626878 3.1996890 14.227008757 18.77600892
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## [30,] 4.939051255 33.961549 -5.6951309 -12.189586982 -12.57568273
## [31,] 6.576476608 50.999889 2.5734551 -12.241210918 -25.07972564
## [32,] 5.606555898 39.844789 -7.4519146 -14.156298736 -16.27396310
## [33,] 5.276745288 42.792539 1.3399468 -11.739840607 -21.43267730
## [34,] 5.467123755 40.678666 -3.1893890 -12.632744705 -18.93103993
## [35,] 6.441721098 42.532845 -2.9622895 -12.490757628 -20.71284634
## [36,] 2.789519263 49.447020 13.9101952 -12.477417569 -19.50643517
## [37,] 7.023293813 39.834628 -6.8494570 -12.398129782 -20.23220353
## [38,] -8.513482253 3.455571 -62.8195719 -13.163582939 -20.17365456
## [39,] 10.839513047 42.715331 33.8210714 -6.860837946 -8.90131640
## [40,] 20.216310932 45.519356 -96.9984894 -25.594820946 -46.81595711
## [41,] 8.748977696 -92.958085 -191.3486036 -5.297592466 -4.18627607
## [42,] 7.919686320 44.010071 -3.6665350 -6.388063703 -10.64961597
## [43,] 10.196584161 55.569076 -6.4180123 -8.544993006 -10.99394295
## [44,] 9.321025810 51.985768 -3.0015351 -7.050787364 -11.62791222
## [45,] 9.697233360 52.411298 -6.5894295 -8.168622248 -10.45551291
## [46,] 8.948412583 51.327946 -6.8684011 -7.726431378 -11.23609623
## [47,] 10.542214998 53.927972 -3.7256335 -8.026235968 -10.36361413
## [48,] 3.099561669 51.995046 -1.3488351 -7.864840821 -10.67674393
## [49,] 23.116736754 66.135329 10.4066011 -7.504250199 -10.89510223
## [50,] -20.789948488 22.872385 -37.1100872 -8.256681388 -10.75128955
## [51,] 34.060739319 14.800029 -34.2698802 -2.711759882 -3.82169791
## [52,] 14.800029459 279.455798 265.5649656 -18.651052361 -27.35996372
## [53,] -34.269880209 265.564966 497.7461031 2.812586044 1.25575294
## [54,] -2.711759882 -18.651052 2.8125860 6.188430786 7.22379802
## [55,] -3.821697912 -27.359964 1.2557529 7.223798024 12.05959148
## [56,] -3.390142508 -23.370063 3.2606136 7.439428186 9.29455576
## [57,] -3.313570628 -24.241047 1.7529066 6.922621916 10.79741877
## [58,] -3.555304996 -24.048590 0.8242367 6.881335110 10.08226031
## [59,] -3.538521758 -22.837513 5.9131735 7.277117213 10.56487575
## [60,] -1.292596230 -36.445853 -22.3496454 6.884022140 10.09022695
## [61,] -4.725347276 3.770043 54.3896571 7.462549337 10.75282328
## [62,] 12.804007615 -8.308461 19.2332810 7.305930770 10.37709022
## [63,] -15.754080672 -4.652863 16.9728980 3.075818945 3.72848411
## [64,] -7.765265412 -130.429987 -117.2976577 15.898230361 26.40459764
## [65,] 13.059489877 -114.563093 -218.5537178 -0.335619971 -0.23865515
## [66,] 0.001243344 0.108348 0.1359816 -0.008509881 -0.01619920
## [,56] [,57] [,58] [,59] [,60]
## [1,] 1.41704989 0.47055267 0.49601537 1.1604498 0.54544734
## [2,] -2.04685621 -1.96505706 -1.81276128 -2.2388056 -1.77516077
## [3,] -1.28501216 0.37743091 0.15829432 -0.6067165 0.06599072
## [4,] -0.69068667 -1.12040407 -0.50221539 -1.4871568 -0.65318758
## [5,] -1.68297878 -0.03978155 -0.57672919 -0.9413140 -0.55744813
## [6,] -1.14944152 -0.88094279 -0.21075556 -1.5514827 -0.30723498
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## [8,] -1.34359606 -0.62541385 -0.45455378 -1.5433080 0.96986867
## [9,] -1.47366230 -0.40999859 -0.50802938 -1.0027874 -1.30314183
## [10,] -1.38444851 -0.48567974 -0.57575386 -1.0814164 -1.01260759
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## [12,] -2.23907166 0.32302290 0.12865899 -1.3925234 0.05707080
## [13,] -1.06769834 -1.09737241 -1.01149377 -1.2402769 -0.88542335
## [14,] -17.67153189 -17.25735414 -16.71217776 -18.1619376 -16.54834486
## [15,] 16.90781355 18.20932520 17.62995144 17.8766243 17.45573754
## [16,] 9.68979579 9.68407299 9.30385038 10.1277069 9.19086733
## [17,] -9.34291949 -9.61126523 -9.37014650 -9.6629632 -9.30962349
## [18,] 16.58203747 16.45239638 15.80886995 17.2668624 15.70608710
## [19,] 19.21750231 17.52037325 17.51511428 18.6606229 17.19709034
## [20,] 15.40656316 17.59970623 15.95035160 17.9930072 16.04424628
## [21,] 18.61100964 17.04448440 17.12445500 18.1273297 16.78650914
## [22,] 16.72761365 17.80598061 15.54270313 18.9433921 16.26907692
## [23,] 18.41562686 16.77641907 17.49941233 17.7734453 16.89212420
## [24,] 17.52862062 17.24593322 16.45073352 17.7845087 19.65827083
## [25,] 17.72575201 17.33297483 16.95252314 18.6374605 12.69949359
## [26,] 17.43639154 17.12332127 16.26665129 17.8945099 18.15473300
## [27,] 9.63878059 9.20840721 8.93246930 10.2089790 5.72228010
## [28,] 37.66617487 37.05119434 36.10231331 38.2924127 38.68322521
## [29,] 4.34562349 4.59322078 4.64952499 3.6347552 11.91590690
## [30,] -13.87280520 -12.29979549 -12.29689970 -13.2435800 -12.38585217
## [31,] -15.97700859 -21.50406461 -19.58665886 -19.9165488 -19.45958377
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## [33,] -15.15224879 -19.66001532 -16.76443159 -18.5774689 -17.20876192
## [34,] -18.51371245 -16.60832374 -19.63998240 -16.0584493 -18.04872049
## [35,] -15.82004514 -19.28775375 -16.17584349 -19.7479864 -15.60277505
## [36,] -17.38335470 -17.55893589 -18.07759833 -16.3858490 -23.19907478
## [37,] -16.43910723 -18.48886420 -17.35313936 -18.7917326 -11.84251590
## [38,] -17.67213173 -18.54236150 -17.50112891 -19.3596724 -12.04481323
## [39,] -8.30911073 -8.35484643 -8.35768529 -8.0646175 -10.94642423
## [40,] -36.98954359 -41.78450341 -40.08837606 -40.2915942 -38.43100510
## [41,] -5.47757751 -4.33767046 -4.25542525 -5.3783100 -0.61761152
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## [43,] -10.53149530 -9.91803407 -9.73838728 -10.5397711 -9.57833618
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## [46,] -8.99209675 -10.28543074 -8.18697697 -10.9849484 -9.11168913
## [47,] -10.26300259 -9.16709858 -10.08278698 -9.6710596 -9.05492544
## [48,] -9.83450272 -9.55995289 -9.15915520 -9.5009639 -14.31988527
## [49,] -9.10572294 -9.70220349 -9.47973723 -10.5778560 -2.06414745
## [50,] -10.04750529 -9.87043577 -8.63647258 -11.0966997 -7.39757518
## [51,] -3.39014251 -3.31357063 -3.55530500 -3.5385218 -1.29259623
## [52,] -23.37006311 -24.24104711 -24.04858963 -22.8375129 -36.44585299
## [53,] 3.26061361 1.75290661 0.82423669 5.9131735 -22.34964536
## [54,] 7.43942819 6.92262192 6.88133511 7.2771172 6.88402214
## [55,] 9.29455576 10.79741877 10.08226031 10.5648758 10.09022695
## [56,] 11.03731988 8.75196083 9.93124456 9.0151597 9.40329428
## [57,] 8.75196083 10.07879317 8.92573170 10.0551814 9.19798300
## [58,] 9.93124456 8.92573170 10.57128167 8.5249173 9.53346520
## [59,] 9.01515974 10.05518142 8.52491730 10.7372683 8.47703363
## [60,] 9.40329428 9.19798300 9.53346520 8.4770336 14.56651504
## [61,] 9.60543584 10.07773319 9.36550357 11.3999981 0.42494503
## [62,] 9.57805395 9.71073619 9.04997141 10.2422196 7.85953781
## [63,] 3.73673235 3.54537883 3.67863402 3.6899809 2.76913514
## [64,] 22.08306082 23.87178007 23.51576035 22.4413326 29.77216355
## [65,] -0.70264131 -0.31231480 0.30383283 -2.3440527 12.02153827
## [66,] -0.01178622 -0.01428646 -0.01388322 -0.0122949 -0.02104195
## [,61] [,62] [,63] [,64] [,65]
## [1,] 1.398485892 1.21046414 0.319353424 0.29783432 -2.80950112
## [2,] -2.519315506 -2.25401047 -0.419379264 -4.85250938 2.52349717
## [3,] -0.820063568 -0.70983594 -0.271416494 2.15412232 3.03638984
## [4,] -1.664448300 -1.29832800 -0.256187050 -1.72651297 2.49206404
## [5,] -1.178312474 -1.17248367 -0.340418547 0.47391176 2.95353856
## [6,] -1.898448023 -1.18406829 -0.402094188 -0.81571736 2.86111320
## [7,] 0.203687910 -1.63568242 0.066440677 -0.24979299 1.99814206
## [8,] -3.523129227 -0.69718839 -0.812262602 0.01096621 4.05794207
## [9,] 0.006643482 -1.82384858 -0.030083392 0.18537587 3.30066582
## [10,] -0.037201541 1.78902337 -1.613234053 -3.18498218 -1.11518576
## [11,] -1.502510999 -2.26367674 0.781671838 -0.10749009 3.95750444
## [12,] -1.986500276 -1.81944091 -0.614716155 1.58345253 2.10950280
## [13,] -1.429042626 -0.91089101 -0.714374360 -7.53886685 -6.26210179
## [14,] -19.166424933 -18.21573252 -6.147114754 -41.45782536 6.24466357
## [15,] 18.299084905 17.81321303 6.401459854 45.30092801 -0.11857121
## [16,] 10.730755715 10.13813401 3.192087799 23.69848309 -3.96736832
## [17,] -9.911897350 -9.65634933 -3.603100900 -23.38242511 0.43674108
## [18,] 18.143782018 17.23654066 5.780562560 39.64783805 -5.68389550
## [19,] 19.865278417 19.00826577 6.450541387 42.20673102 -6.89088737
## [20,] 18.693391490 17.49403634 5.870384867 41.93658395 -5.13773343
## [21,] 19.332853854 18.56112659 6.243424669 41.10686740 -6.90194958
## [22,] 19.337041997 18.51877724 5.924633316 41.53758042 -6.36586266
## [23,] 18.760341552 17.91389070 6.374327232 41.26232123 -6.01770489
## [24,] 14.311337245 20.68013751 4.399047131 41.79427080 -4.22448202
## [25,] 25.008866540 13.86928890 8.980091048 41.56259891 -8.40079806
## [26,] 17.674005663 27.31402948 0.054393109 40.19993084 -3.30580031
## [27,] 16.049921653 6.88631315 10.029456694 14.56476027 -20.96013994
## [28,] 33.975348363 35.66709581 11.613736679 103.36416120 11.98167297
## [29,] -10.846214804 1.24035376 -5.664823668 40.08575353 53.00660566
## [30,] -13.529636211 -13.28842582 -5.841817358 -27.58399387 0.51930100
## [31,] -20.202225424 -19.33027025 -6.487587958 -54.29770498 -1.04632674
## [32,] -17.767938844 -17.98579242 -6.908321882 -40.07110769 1.90842344
## [33,] -18.531175092 -17.59890843 -6.179238275 -47.68927291 -1.06477201
## [34,] -17.414017759 -17.59989108 -6.444578529 -43.78893668 0.54493718
## [35,] -20.777791819 -17.48220945 -6.816141465 -45.78908120 0.87669277
## [36,] -9.352435594 -17.68918632 -5.247668504 -48.89222936 -8.06812038
## [37,] -29.781261206 -16.89772694 -7.369506969 -44.17273963 2.99766431
## [38,] -32.613475469 -38.09796681 1.398106574 -23.40911740 33.16529149
## [39,] -2.660225007 0.17723508 -9.146817970 -31.06563512 -16.86337213
## [40,] -43.735050148 -37.81076238 -15.974200389 -90.26163628 33.67233577
## [41,] -13.534959103 -7.34713796 -4.224178634 29.96119654 66.39803276
## [42,] -10.062365696 -9.53571556 -2.774104162 -23.94428489 3.24068641
## [43,] -11.230622242 -10.63760037 -3.440035957 -24.11782718 4.36568526
## [44,] -10.373715232 -9.68898232 -2.932576981 -24.79954358 3.03204762
## [45,] -10.893896494 -10.30871621 -3.320370219 -23.17841460 4.41143068
## [46,] -10.930954856 -10.90979317 -2.877259424 -23.50665792 4.83865638
## [47,] -10.770754178 -9.08420805 -3.684150216 -23.99252085 2.88147323
## [48,] -3.086499954 -14.70141554 -0.366058479 -23.11682158 2.66625403
## [49,] -20.398161994 2.93127080 -9.225900346 -31.87454987 -5.95821008
## [50,] -20.189383766 -38.69883447 11.245961785 -5.61344818 24.84871383
## [51,] -4.725347276 12.80400762 -15.754080672 -7.76526541 13.05948988
## [52,] 3.770042878 -8.30846081 -4.652862731 -130.42998663 -114.56309259
## [53,] 54.389657108 19.23328100 16.972897974 -117.29765768 -218.55371779
## [54,] 7.462549337 7.30593077 3.075818945 15.89823036 -0.33561997
## [55,] 10.752823281 10.37709022 3.728484106 26.40459764 -0.23865515
## [56,] 9.605435842 9.57805395 3.736732353 22.08306082 -0.70264131
## [57,] 10.077733186 9.71073619 3.545378830 23.87178007 -0.31231480
## [58,] 9.365503566 9.04997141 3.678634019 23.51576035 0.30383283
## [59,] 11.399998127 10.24221955 3.689980878 22.44133263 -2.34405271
## [60,] 0.424945034 7.85953781 2.769135144 29.77216355 12.02153827
## [61,] 28.862916529 15.68335028 4.132874537 8.72821310 -27.16498763
## [62,] 15.683350277 25.22233704 -4.397237345 13.45802920 -11.87854158
## [63,] 4.132874537 -4.39723735 9.162749329 7.72351886 -5.71824401
## [64,] 8.728213095 13.45802920 7.723518858 92.82139968 55.10907976
## [65,] -27.164987626 -11.87854157 -5.718244014 55.10907976 101.76728497
## [66,] 0.003386987 -0.00252104 -0.003289057 -0.07267699 -0.06354741
## [,66]
## [1,] 0.0081103366
## [2,] -0.0033054483
## [3,] -0.0108417785
## [4,] -0.0061453221
## [5,] -0.0091968864
## [6,] -0.0074560466
## [7,] -0.0082288412
## [8,] -0.0084713360
## [9,] -0.0087623221
## [10,] -0.0047372114
## [11,] -0.0049553960
## [12,] -0.0183292626
## [13,] 0.0046567713
## [14,] 0.0177716929
## [15,] -0.0314170618
## [16,] -0.0108620200
## [17,] 0.0135729154
## [18,] -0.0164816472
## [19,] -0.0174713145
## [20,] -0.0194526259
## [21,] -0.0169603788
## [22,] -0.0181289659
## [23,] -0.0172400804
## [24,] -0.0182767013
## [25,] -0.0177667001
## [26,] -0.0167370821
## [27,] 0.0005567313
## [28,] -0.0557844685
## [29,] -0.0374930898
## [30,] 0.0168958609
## [31,] 0.0405990919
## [32,] 0.0239612917
## [33,] 0.0350247971
## [34,] 0.0294879294
## [35,] 0.0320577097
## [36,] 0.0355357775
## [37,] 0.0305716202
## [38,] 0.0061519236
## [39,] 0.0254817563
## [40,] 0.0568310686
## [41,] -0.0388843688
## [42,] 0.0117231375
## [43,] 0.0106306073
## [44,] 0.0129780228
## [45,] 0.0098866941
## [46,] 0.0108102803
## [47,] 0.0109510604
## [48,] 0.0102627875
## [49,] 0.0203639287
## [50,] -0.0092099291
## [51,] 0.0012433444
## [52,] 0.1083480123
## [53,] 0.1359816294
## [54,] -0.0085098810
## [55,] -0.0161991982
## [56,] -0.0117862219
## [57,] -0.0142864643
## [58,] -0.0138832192
## [59,] -0.0122949006
## [60,] -0.0210419472
## [61,] 0.0033869869
## [62,] -0.0025210397
## [63,] -0.0032890570
## [64,] -0.0726769904
## [65,] -0.0635474062
## [66,] 0.0050834145
##
## $log_evidence
## [1] -367.8081
##
## $converge
## [1] "YES"
##
## $iter_counts
## [1] 233
A function is defined for you in the code chunk below. This function
creates a coefficient summary plot in the style of the
coefplot() function, but uses the Bayesian results from the
Laplace Approximation. The first argument is the vector of posterior
means, and the second argument is the vector of posterior standard
deviations. The third argument is the name of the feature associated
with each coefficient.
viz_post_coefs <- function(post_means, post_sds, xnames)
{
tibble::tibble(
mu = post_means,
sd = post_sds,
x = xnames
) %>%
mutate(x = factor(x, levels = xnames)) %>%
ggplot(mapping = aes(x = x)) +
geom_hline(yintercept = 0, color = 'grey', linetype = 'dashed') +
geom_point(mapping = aes(y = mu)) +
geom_linerange(mapping = aes(ymin = mu - 2 * sd,
ymax = mu + 2 * sd,
group = x)) +
labs(x = 'feature', y = 'coefficient value') +
coord_flip() +
theme_bw()
}
Create the posterior summary visualization figure for model 3
and model 6. You must provide the posterior means and standard
deviations of the regression coefficients (the \(\beta\) parameters). Do NOT include the
\(\varphi\) parameter. The feature
names associated with the coefficients can be extracted from the design
matrix using the colnames() function.
### make the posterior coefficient visualization for model 3
viz_post_coefs(laplace_03_weak$mode[-length(laplace_03_weak$mode)],
sqrt(diag(laplace_03_weak$var_matrix))[-length(laplace_03_weak$mode)],
info_03_weak$design_matrix %>% colnames())
### make the posterior coefficient visualization for model 6
viz_post_coefs(laplace_06_weak$mode[-length(laplace_06_weak$mode)],
sqrt(diag(laplace_06_weak$var_matrix))[-length(laplace_06_weak$mode)],
info_06_weak$design_matrix %>% colnames())
Use the Bayes Factor to identify the better of the models.
### add more code chunks if you like
log_evidences <- c(laplace_03_weak$log_evidence, laplace_06_weak$log_evidence)
if (exp(log_evidences[1]) > exp(log_evidences[2])) {
sprintf("Model 3 is better and has bayes factor of:%f", exp(log_evidences[1])/sum(exp(log_evidences)))
} else {
sprintf("Model 6 is better and has bayes factor of:%f", exp(log_evidences[2])/sum(exp(log_evidences)))
}
## [1] "Model 3 is better and has bayes factor of:1.000000"
You fit the Bayesian models assuming a diffuse or weak prior. Let’s now try a more informative or strong prior by reducing the prior standard deviation on the regression coefficients from 50 to 1. The prior mean will still be zero.
Complete the first code chunk below, which defines the list of required information for both the model 3 and model 6 formulations using the strong prior on the regression coefficients. All other information, data and the \(\sigma\) prior, are the same as before.
Run the Laplace Approximation using the strong prior for both
the model 3 and model 6 formulations. Assign the results to
laplace_03_strong and
laplace_06_strong.
Confirm that the optimizations converged for both laplace approximation results.
Define the lists of required information for the strong prior.
info_03_strong <- list(
yobs = df$y,
design_matrix = model.matrix(y ~ (x1 + I(x1^2))*(x2 + I(x2^2)), data = df),
mu_beta = 0,
tau_beta = 1,
sigma_rate = 1
)
info_06_strong <- list(
yobs = df$y,
design_matrix = model.matrix(y ~ splines::ns(x1, df = 12) * (x2 +I(x2^2) + I(x2^3) + I(x2^4)), data = df),
mu_beta = 0,
tau_beta = 1,
sigma_rate = 1
)
Execute the Laplace Approximation.
### add more code chunks if you like
laplace_03_strong <- my_laplace(start_guess_03,
lm_logpost,
info_03_strong)
laplace_03_strong
## $mode
## [1] 0.651631489 0.162914437 -0.152730088 -0.051739385 -0.548524155
## [6] 0.118435267 -0.079153776 0.002640972 0.017130016 -0.398726740
##
## $var_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.0135459998 -2.873370e-04 -6.802265e-03 3.977485e-04 -5.671532e-03
## [2,] -0.0002873370 7.716907e-03 1.696332e-04 7.896033e-04 -5.465947e-04
## [3,] -0.0068022649 1.696332e-04 7.455503e-03 1.562543e-04 3.314492e-03
## [4,] 0.0003977485 7.896033e-04 1.562543e-04 8.086078e-03 5.740636e-04
## [5,] -0.0056715321 -5.465947e-04 3.314492e-03 5.740636e-04 5.582248e-03
## [6,] 0.0023593465 -3.371931e-05 1.560391e-04 5.950152e-04 -1.881054e-03
## [7,] -0.0016154477 -2.928150e-03 5.844267e-04 -1.063987e-03 1.486913e-03
## [8,] 0.0012803561 1.764095e-04 -5.707103e-04 -2.435585e-03 -1.561461e-03
## [9,] 0.0024573094 3.145161e-04 -3.430520e-03 -8.954372e-04 -2.355658e-03
## [10,] -0.0001336134 -1.536557e-05 7.457318e-05 2.363612e-06 7.767143e-05
## [,6] [,7] [,8] [,9] [,10]
## [1,] 2.359346e-03 -0.0016154477 0.0012803561 2.457309e-03 -1.336134e-04
## [2,] -3.371931e-05 -0.0029281504 0.0001764095 3.145161e-04 -1.536557e-05
## [3,] 1.560391e-04 0.0005844267 -0.0005707103 -3.430520e-03 7.457318e-05
## [4,] 5.950152e-04 -0.0010639870 -0.0024355848 -8.954372e-04 2.363612e-06
## [5,] -1.881054e-03 0.0014869130 -0.0015614607 -2.355658e-03 7.767143e-05
## [6,] 1.197126e-02 -0.0044269068 0.0041668287 -2.733656e-03 -4.241310e-05
## [7,] -4.426907e-03 0.0049132924 -0.0025030729 8.204208e-04 3.285110e-05
## [8,] 4.166829e-03 -0.0025030729 0.0044113495 -4.354029e-04 -2.628410e-05
## [9,] -2.733656e-03 0.0008204208 -0.0004354029 3.417039e-03 -3.183327e-05
## [10,] -4.241310e-05 0.0000328511 -0.0000262841 -3.183327e-05 5.001177e-03
##
## $log_evidence
## [1] -130.025
##
## $converge
## [1] "YES"
##
## $iter_counts
## [1] 112
laplace_06_strong <- my_laplace(start_guess_06,
lm_logpost,
info_06_strong)
laplace_06_strong
## $mode
## [1] 0.3212233451 -0.0287175403 -0.1004156846 0.0563850943 0.7265367354
## [6] 0.6241984921 0.1763432345 0.2157951086 0.3251316206 -0.2593082249
## [11] 0.8798067880 0.1410587249 -0.0375962861 -0.2760227845 -0.5765233744
## [16] -0.0787870208 -0.1935412001 0.4183144344 -0.1790317991 -0.1461579133
## [21] 0.3161817413 0.4519048156 0.7715572767 -0.4720424034 -0.6946505909
## [26] 0.9825140738 -0.0213411466 -0.0923919177 0.2747168212 0.4445480819
## [31] -0.1194984204 0.4518601321 0.1670873690 0.1368712616 -0.2049106757
## [36] -0.2347101877 0.1132589673 0.0004572422 -0.0504898864 0.3036721392
## [41] -0.0703075815 -0.1455324735 0.3850220230 -0.0871126242 0.0073550637
## [46] -0.3653392780 0.2061344063 0.6149856026 -0.0792086730 0.4087060682
## [51] 0.0046394853 0.0792319406 0.1491674989 0.1128291515 0.1628276525
## [56] 0.5346678717 -0.1516368992 0.2565965590 0.0342450451 0.1988417877
## [61] -0.0787726228 0.0736476716 0.1600793707 0.3558464821 -0.0149379291
## [66] -0.5995065976
##
## $var_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 7.388224e-02 -0.0600681013 -0.0700384724 -6.318657e-02 -0.0692514124
## [2,] -6.006810e-02 0.3030111866 -0.0522872815 9.295836e-02 0.0421116557
## [3,] -7.003847e-02 -0.0522872815 0.2342916330 -1.080294e-02 0.0966653317
## [4,] -6.318657e-02 0.0929583563 -0.0108029361 2.575281e-01 -0.0326693234
## [5,] -6.925141e-02 0.0421116557 0.0966653317 -3.266932e-02 0.2391828267
## [6,] -6.188450e-02 0.0601917161 0.0460860172 9.917915e-02 -0.0256958634
## [7,] -5.925645e-02 0.0471795192 0.0624238493 3.886433e-02 0.0824027667
## [8,] -6.645340e-02 0.0579107866 0.0632416563 6.225290e-02 0.0576272145
## [9,] -6.680050e-02 0.0549305810 0.0646731466 5.578354e-02 0.0664241818
## [10,] -6.275211e-02 0.0543957378 0.0610999891 5.587478e-02 0.0599830225
## [11,] -4.630043e-02 0.0781851973 0.0243305942 4.552325e-02 0.0412837316
## [12,] -1.202120e-01 0.0020207454 0.1647911510 8.962006e-02 0.1235539896
## [13,] -1.887969e-03 0.0606090479 -0.0249700718 1.286167e-02 -0.0015117337
## [14,] -1.550560e-03 0.0124370826 0.0009073550 -4.000785e-03 0.0041239914
## [15,] -2.782347e-02 0.0098213878 0.0149969212 1.459756e-02 0.0124783230
## [16,] 4.094768e-03 -0.0049919506 -0.0052256016 -2.711798e-05 -0.0057922466
## [17,] -1.194747e-03 0.0109292630 0.0048035247 6.511931e-03 0.0074201188
## [18,] 9.086165e-03 -0.0153093590 0.0108009577 -7.697452e-03 -0.0095016989
## [19,] -2.827811e-03 0.0005823317 0.0298071030 -2.971565e-02 0.0200130649
## [20,] -3.680784e-03 -0.0029753677 -0.0109434612 -2.754010e-02 -0.0114300334
## [21,] 9.905912e-04 -0.0172679139 0.0128424858 -2.229154e-02 0.0247058699
## [22,] 1.124637e-03 -0.0084113607 -0.0034435675 1.035762e-02 -0.0095620917
## [23,] -2.047138e-05 -0.0077619854 0.0024672678 2.490237e-03 -0.0001262920
## [24,] 2.638240e-03 -0.0102166776 -0.0035820355 6.224214e-03 -0.0108061411
## [25,] 1.425452e-03 -0.0095139101 -0.0019130625 9.606913e-04 -0.0040557053
## [26,] 2.767152e-03 -0.0089495712 -0.0002779557 1.353545e-03 -0.0015901109
## [27,] 1.155304e-02 -0.0197241838 -0.0013991342 -9.570576e-03 -0.0089008510
## [28,] -1.114963e-02 0.0229455662 -0.0043476084 2.410593e-02 0.0091313209
## [29,] 3.493702e-03 -0.0153776881 0.0094426847 -5.883419e-03 0.0003000915
## [30,] 4.593264e-03 -0.1619183584 0.0367043535 -9.229096e-03 0.0090283630
## [31,] 4.485249e-03 0.0230946573 -0.1014358021 -1.408500e-02 0.0067107654
## [32,] 2.937857e-03 0.0160789042 -0.0048570093 -1.012573e-01 0.0035761798
## [33,] 1.044540e-02 -0.0002613912 -0.0132938279 3.704902e-02 -0.1437520392
## [34,] 1.164653e-02 -0.0026999051 -0.0060279160 -2.680409e-03 0.0177468264
## [35,] 9.633165e-03 -0.0018545111 0.0016260669 -1.128567e-02 0.0106749561
## [36,] 8.234431e-03 -0.0012174804 -0.0031510469 -4.632151e-04 -0.0040580096
## [37,] 1.093229e-03 0.0009841793 0.0015899668 -4.684053e-04 0.0029220415
## [38,] 1.082593e-02 -0.0031613652 -0.0038847634 -3.791794e-03 -0.0033756462
## [39,] 1.431648e-02 -0.0065692970 -0.0125575695 -8.660879e-03 -0.0096651076
## [40,] -8.427393e-03 0.0037433393 0.0209079351 1.107838e-02 0.0153559020
## [41,] -7.156218e-03 0.0068308752 0.0026213862 6.339391e-03 0.0053641775
## [42,] 4.902306e-03 -0.0037615938 0.0069189911 -1.888200e-03 -0.0025439472
## [43,] -1.008890e-02 0.0014170970 -0.0060749219 6.190936e-03 0.0058901702
## [44,] -2.399836e-03 -0.0064290921 0.0126928159 1.846267e-02 -0.0027625487
## [45,] -7.663466e-04 0.0078585955 -0.0002287436 1.404492e-03 0.0090972442
## [46,] -5.395781e-03 0.0050031235 0.0029050688 7.664331e-03 -0.0042367890
## [47,] -2.030070e-03 0.0017629535 0.0060188433 -6.714261e-03 0.0142903740
## [48,] -9.234420e-03 0.0052732048 0.0090208998 3.281950e-03 0.0098011625
## [49,] -3.520233e-03 0.0010156129 0.0032129542 -4.242595e-04 0.0031102082
## [50,] -6.783795e-03 -0.0014609575 0.0047945284 2.922520e-03 0.0028333876
## [51,] -2.364585e-04 -0.0034573613 -0.0029224820 5.355254e-06 -0.0036535825
## [52,] -2.767184e-03 -0.0029089369 0.0084403319 -3.784862e-03 0.0055735956
## [53,] 3.821541e-03 -0.0039456156 -0.0052043988 -3.496736e-04 -0.0046627554
## [54,] 4.294554e-03 -0.0025510122 -0.0050286015 -8.162264e-03 -0.0087720505
## [55,] 6.441789e-03 -0.0060657289 0.0049224436 -1.788830e-03 -0.0181082718
## [56,] 1.337726e-02 -0.0239182572 0.0015898082 -5.216645e-03 0.0119449599
## [57,] 3.003508e-03 -0.0115227812 -0.0084596966 -8.145852e-03 0.0070702220
## [58,] 8.684737e-03 -0.0119222143 -0.0061553781 -1.740817e-02 0.0017581762
## [59,] 2.800952e-03 -0.0126099530 -0.0084590441 -2.571806e-03 -0.0231336044
## [60,] 9.762911e-03 -0.0143443970 -0.0097046036 -1.275550e-02 -0.0101778200
## [61,] 1.040481e-02 -0.0115929275 -0.0094981683 -1.184073e-02 -0.0101530516
## [62,] 5.727726e-03 -0.0093742710 -0.0065709680 -9.369958e-03 -0.0078859861
## [63,] -3.419063e-03 -0.0089434166 0.0062413979 -1.684408e-03 0.0019287352
## [64,] 2.096018e-02 -0.0085640365 -0.0323566992 -2.398399e-02 -0.0266084112
## [65,] 3.528548e-04 -0.0113283603 0.0067236209 -2.681536e-03 0.0012885121
## [66,] 1.158708e-03 -0.0001972937 -0.0015276575 -4.302898e-04 -0.0019512122
## [,6] [,7] [,8] [,9] [,10]
## [1,] -0.0618844995 -5.925645e-02 -0.0664533976 -0.0668004993 -0.0627521099
## [2,] 0.0601917161 4.717952e-02 0.0579107866 0.0549305810 0.0543957378
## [3,] 0.0460860172 6.242385e-02 0.0632416563 0.0646731466 0.0610999891
## [4,] 0.0991791472 3.886433e-02 0.0622528959 0.0557835430 0.0558747751
## [5,] -0.0256958634 8.240277e-02 0.0576272145 0.0664241818 0.0599830225
## [6,] 0.2843430685 -2.680902e-02 0.0792400982 0.0485566828 0.0599659814
## [7,] -0.0268090151 3.358141e-01 -0.0424126902 0.0935445981 0.0335976685
## [8,] 0.0792400982 -4.241269e-02 0.2434552860 -0.0090303616 0.0925857382
## [9,] 0.0485566828 9.354460e-02 -0.0090303616 0.2401918620 -0.0343100525
## [10,] 0.0599659814 3.359767e-02 0.0925857382 -0.0343100525 0.3010958475
## [11,] 0.0377478608 4.536824e-02 0.0266351882 0.0825084646 -0.0698418099
## [12,] 0.1023341103 1.045175e-01 0.1124274543 0.1162886379 0.0937926773
## [13,] 0.0054659942 -2.319152e-03 0.0107851012 -0.0178497444 0.0540533253
## [14,] 0.0053374422 -6.206362e-03 0.0108385393 0.0031806247 0.0089942874
## [15,] 0.0071109322 6.727562e-03 0.0050703507 0.0177403804 0.0024876655
## [16,] -0.0062792546 6.902896e-05 -0.0093113674 -0.0046721281 -0.0082702415
## [17,] 0.0084636506 8.482114e-03 0.0107490866 0.0044297128 0.0112035864
## [18,] -0.0111975494 -1.557268e-03 -0.0158541323 -0.0093686753 -0.0143549788
## [19,] -0.0120825013 1.158507e-02 -0.0085830587 0.0009282996 -0.0056279819
## [20,] 0.0057565980 4.739539e-03 -0.0020088324 0.0011866409 -0.0023776218
## [21,] -0.0105548402 8.035785e-03 -0.0127841342 -0.0009330031 -0.0078869284
## [22,] 0.0166623605 -9.152857e-03 -0.0005199630 -0.0046244561 -0.0062778312
## [23,] 0.0029364951 -2.477981e-02 -0.0165024593 0.0052104601 -0.0010532986
## [24,] 0.0053273644 -5.140995e-02 0.0730009407 -0.0152753918 -0.0125677258
## [25,] -0.0053449889 3.340059e-03 -0.0114210684 0.0311337425 -0.0125742491
## [26,] -0.0027140856 8.329672e-03 -0.0087221226 -0.0193556733 0.0987053449
## [27,] -0.0081739379 -9.087774e-03 -0.0032847085 -0.0162965625 -0.0088092767
## [28,] 0.0101569521 2.207111e-02 0.0052140419 0.0134730938 -0.0094599783
## [29,] -0.0021555264 2.679413e-03 -0.0047344491 0.0085630785 -0.0280914594
## [30,] 0.0029186075 6.944260e-03 0.0049704428 0.0001276787 0.0063957015
## [31,] -0.0037197323 -3.634526e-03 0.0060218490 -0.0012705412 0.0052388762
## [32,] 0.0042803388 -1.327527e-03 0.0058419516 -0.0004914935 0.0048047717
## [33,] 0.0215279863 2.371236e-03 -0.0039568080 -0.0019915947 0.0008126642
## [34,] -0.1384300581 -7.193283e-03 0.0196557429 -0.0135815109 0.0074335223
## [35,] 0.0024036094 -1.595216e-01 0.0311265614 -0.0093338021 0.0053277888
## [36,] 0.0031159629 2.193105e-02 -0.1574426795 0.0170328667 0.0001583417
## [37,] -0.0025220752 1.802153e-02 -0.0040405982 -0.1203510718 0.0137242344
## [38,] 0.0010058367 -2.453537e-03 -0.0036658425 0.0135640518 -0.1169015851
## [39,] -0.0088995547 -1.557394e-03 -0.0153604403 0.0074083248 -0.0240260288
## [40,] 0.0141431937 1.972769e-02 0.0122584054 0.0197116388 0.0012933115
## [41,] 0.0041506954 5.443443e-03 0.0049529477 0.0070002297 -0.0083788455
## [42,] 0.0023587695 -3.818267e-03 0.0048398081 -0.0023487351 0.0045658150
## [43,] 0.0107924819 3.118079e-03 0.0112551181 0.0091369153 0.0100107699
## [44,] 0.0029372379 -1.384123e-02 0.0108639823 0.0016206451 0.0057058802
## [45,] 0.0088585513 5.642243e-03 0.0061588453 0.0026977396 0.0074822064
## [46,] -0.0169021762 1.376486e-02 0.0111647400 0.0030303034 0.0073406435
## [47,] 0.0080231452 1.436297e-02 0.0016509927 0.0049870597 0.0029090000
## [48,] 0.0041298556 2.250090e-02 -0.0201310705 -0.0100192511 0.0162104859
## [49,] 0.0032938360 -1.633667e-03 -0.0186461991 0.0272996454 -0.0149281682
## [50,] 0.0024708362 1.563866e-03 -0.0016969886 0.0042295336 0.0032242131
## [51,] -0.0025859767 -5.435892e-03 -0.0014378770 -0.0028543752 -0.0231738328
## [52,] 0.0045710181 -9.245620e-04 0.0084097174 0.0038813864 -0.0031713167
## [53,] -0.0041730747 4.473532e-04 -0.0074911824 -0.0012972374 -0.0097531565
## [54,] -0.0096883385 -8.844492e-03 -0.0114954856 -0.0066476782 -0.0115796947
## [55,] -0.0075183641 -1.221425e-02 -0.0127563604 -0.0083237355 -0.0131201275
## [56,] -0.0201079366 1.903333e-03 -0.0183787069 -0.0119988180 -0.0140774929
## [57,] -0.0055876135 -2.028674e-02 -0.0077161968 -0.0073193020 -0.0113459168
## [58,] -0.0049297443 1.853126e-02 -0.0221427199 -0.0072827997 -0.0147321171
## [59,] 0.0118014677 -7.632775e-03 -0.0057538018 -0.0089842000 -0.0089623848
## [60,] -0.0148932648 -8.273551e-03 0.0203518922 0.0051733678 -0.0162953866
## [61,] -0.0110789158 -1.280121e-02 0.0151154185 -0.0149850240 -0.0028197219
## [62,] -0.0078355277 -9.636434e-03 -0.0086586454 -0.0013658672 -0.0074101978
## [63,] 0.0000880042 -3.335098e-04 0.0021410025 -0.0049521897 0.0182957630
## [64,] -0.0233362959 -2.830469e-02 -0.0263816842 -0.0254259202 -0.0111545846
## [65,] 0.0024200786 -3.141482e-03 0.0025386953 -0.0009411538 0.0163542002
## [66,] -0.0017377743 -1.939218e-03 -0.0005929875 -0.0016450490 -0.0003281460
## [,11] [,12] [,13] [,14] [,15]
## [1,] -0.0463004273 -0.120211972 -1.887969e-03 -0.0015505603 -0.0278234736
## [2,] 0.0781851973 0.002020745 6.060905e-02 0.0124370826 0.0098213878
## [3,] 0.0243305942 0.164791151 -2.497007e-02 0.0009073550 0.0149969212
## [4,] 0.0455232499 0.089620058 1.286167e-02 -0.0040007854 0.0145975582
## [5,] 0.0412837316 0.123553990 -1.511734e-03 0.0041239914 0.0124783230
## [6,] 0.0377478608 0.102334110 5.465994e-03 0.0053374422 0.0071109322
## [7,] 0.0453682382 0.104517452 -2.319152e-03 -0.0062063622 0.0067275618
## [8,] 0.0266351882 0.112427454 1.078510e-02 0.0108385393 0.0050703507
## [9,] 0.0825084646 0.116288638 -1.784974e-02 0.0031806247 0.0177403804
## [10,] -0.0698418099 0.093792677 5.405333e-02 0.0089942874 0.0024876655
## [11,] 0.2416663520 0.013249780 -8.640244e-02 0.0043420986 0.0219564846
## [12,] 0.0132497795 0.450201427 5.776854e-02 -0.0067404439 0.0046553333
## [13,] -0.0864024398 0.057768543 3.824981e-01 -0.0002534069 -0.0074295011
## [14,] 0.0043420986 -0.006740444 -2.534069e-04 0.0859650645 -0.0082750507
## [15,] 0.0219564846 0.004655333 -7.429501e-03 -0.0082750507 0.1267424465
## [16,] -0.0022426453 -0.006367061 -1.990814e-03 -0.0293028348 0.0113354128
## [17,] -0.0008721441 0.020087810 3.200616e-03 0.0113532290 -0.0486902397
## [18,] -0.0138946082 0.008574603 -9.760842e-03 -0.0723831118 0.0014941560
## [19,] 0.0032917534 -0.003097846 5.676784e-03 -0.0637735948 0.0209357955
## [20,] -0.0011316288 0.012780409 -1.207074e-03 -0.0558134148 0.0116897266
## [21,] -0.0038931350 0.008095171 -1.226051e-03 -0.0733342168 0.0077739136
## [22,] -0.0028266023 0.004869116 4.925614e-05 -0.0646531073 0.0049324663
## [23,] -0.0040310596 0.012084163 1.944954e-03 -0.0596174850 -0.0079446962
## [24,] -0.0025448441 0.003513660 -1.233001e-03 -0.0688709420 0.0051271832
## [25,] -0.0058487753 0.003100137 6.420611e-03 -0.0558239321 0.0157941997
## [26,] -0.0340052573 -0.007987502 -2.962672e-03 -0.0556029090 -0.0122762870
## [27,] -0.0309799420 0.018986677 3.613291e-02 -0.0184613091 -0.0195820650
## [28,] 0.0321253121 -0.003025064 -2.186467e-02 -0.1007316314 -0.0206003098
## [29,] 0.0374506698 -0.025204235 -1.108706e-01 0.0030964330 -0.0155605885
## [30,] -0.0053939864 0.015552799 1.294462e-03 0.0055627539 -0.0608850550
## [31,] -0.0029944965 0.002718078 6.279973e-03 0.0128118910 -0.0477656194
## [32,] 0.0010476937 -0.001102025 7.304740e-03 0.0074101521 -0.0402089258
## [33,] -0.0085934497 0.006152593 3.132205e-03 0.0072945526 -0.0678388840
## [34,] -0.0132798654 0.008578665 5.115408e-03 0.0023920351 -0.0788821200
## [35,] -0.0094172069 0.011658128 5.331548e-03 -0.0006092628 -0.0774130187
## [36,] -0.0078602878 0.005379584 4.408650e-03 -0.0036522306 -0.0548535820
## [37,] 0.0000804220 0.005644967 6.421238e-04 0.0035082619 -0.0228266312
## [38,] -0.0072144930 0.008585064 9.106021e-03 -0.0019547449 -0.0672227179
## [39,] -0.1172718790 0.027128367 2.998261e-02 -0.0017470552 -0.0379104819
## [40,] 0.0360261403 -0.112521214 -5.914378e-03 -0.0247463941 -0.0615965514
## [41,] 0.0256608120 -0.004501557 -1.973703e-01 -0.0036784659 0.0131160500
## [42,] -0.0138128212 0.022513240 -8.497264e-03 0.0229421640 -0.0377273136
## [43,] 0.0084578950 0.006553564 3.296117e-03 0.0246423003 0.0073635222
## [44,] -0.0018403275 0.010767822 -1.168568e-03 0.0148523490 -0.0103747110
## [45,] 0.0009136342 0.002129096 4.586533e-03 0.0278851731 -0.0240587188
## [46,] 0.0020160696 0.008019648 1.777476e-03 0.0146158745 -0.0027476912
## [47,] 0.0022566350 0.002129393 2.138568e-03 0.0180893930 -0.0072691130
## [48,] 0.0042759070 0.008960151 -6.988548e-04 0.0136571212 0.0073141001
## [49,] 0.0104506381 0.006850145 4.017773e-03 0.0098287613 0.0078064506
## [50,] 0.0066261156 -0.003787253 -1.423074e-02 -0.0139572045 0.0268347644
## [51,] 0.0017895006 -0.001090782 1.484821e-02 -0.0144453511 0.0205640979
## [52,] 0.0065478130 0.008358124 -2.077287e-02 0.0137335752 -0.0112652160
## [53,] 0.0166409563 -0.027009025 -4.007133e-02 -0.0203988764 -0.0024847277
## [54,] -0.0060094586 -0.010352882 -7.555793e-03 -0.0146406521 0.0357976537
## [55,] -0.0017450861 -0.026325346 -9.902677e-04 -0.0102226959 0.0324782594
## [56,] -0.0095887018 -0.021963409 -2.010189e-03 -0.0114688384 0.0018710024
## [57,] -0.0009120609 -0.021347410 -2.891795e-03 -0.0116544096 0.0440419804
## [58,] -0.0029671368 -0.026424028 -2.418905e-04 -0.0136186439 0.0221775652
## [59,] -0.0014373738 -0.019370235 -3.812127e-03 -0.0007215817 0.0415663433
## [60,] -0.0073458634 -0.025967848 -3.332973e-03 -0.0059462236 0.0155785247
## [61,] -0.0016953214 -0.014597513 6.413762e-03 0.0043906048 0.0003722837
## [62,] 0.0052562254 -0.016301979 -1.736189e-03 0.0072477418 0.0176506554
## [63,] 0.0159555744 -0.011284878 -1.017452e-02 0.0052925822 0.0190139896
## [64,] -0.0209816906 -0.030288274 9.937924e-03 0.0045919993 0.0347583997
## [65,] -0.0122245356 0.013875300 5.171820e-02 0.0104689558 -0.0104446211
## [66,] -0.0028224623 -0.001529064 1.338725e-03 0.0009945125 0.0006510796
## [,16] [,17] [,18] [,19] [,20]
## [1,] 4.094768e-03 -0.0011947474 0.0090861650 -0.0028278106 -0.0036807843
## [2,] -4.991951e-03 0.0109292630 -0.0153093590 0.0005823317 -0.0029753677
## [3,] -5.225602e-03 0.0048035247 0.0108009577 0.0298071030 -0.0109434612
## [4,] -2.711798e-05 0.0065119310 -0.0076974524 -0.0297156511 -0.0275401021
## [5,] -5.792247e-03 0.0074201188 -0.0095016989 0.0200130649 -0.0114300334
## [6,] -6.279255e-03 0.0084636506 -0.0111975494 -0.0120825013 0.0057565980
## [7,] 6.902896e-05 0.0084821137 -0.0015572680 0.0115850661 0.0047395391
## [8,] -9.311367e-03 0.0107490866 -0.0158541323 -0.0085830587 -0.0020088324
## [9,] -4.672128e-03 0.0044297128 -0.0093686753 0.0009282996 0.0011866409
## [10,] -8.270242e-03 0.0112035864 -0.0143549788 -0.0056279819 -0.0023776218
## [11,] -2.242645e-03 -0.0008721441 -0.0138946082 0.0032917534 -0.0011316288
## [12,] -6.367061e-03 0.0200878103 0.0085746031 -0.0030978463 0.0127804095
## [13,] -1.990814e-03 0.0032006160 -0.0097608419 0.0056767835 -0.0012070739
## [14,] -2.930283e-02 0.0113532290 -0.0723831118 -0.0637735948 -0.0558134148
## [15,] 1.133541e-02 -0.0486902397 0.0014941560 0.0209357955 0.0116897266
## [16,] 5.891249e-02 0.0170854336 0.0234830580 0.0157188459 0.0149857459
## [17,] 1.708543e-02 0.0512074770 -0.0144368136 -0.0178860112 -0.0119570177
## [18,] 2.348306e-02 -0.0144368136 0.4801862601 -0.0585122227 0.0654221228
## [19,] 1.571885e-02 -0.0178860112 -0.0585122227 0.4844140003 -0.0900967092
## [20,] 1.498575e-02 -0.0119570177 0.0654221228 -0.0900967092 0.5626672032
## [21,] 2.156564e-02 -0.0121960171 0.0602407700 0.0824856890 -0.0646324792
## [22,] 1.584630e-02 -0.0112439757 0.0560224146 0.0402287242 0.0695429368
## [23,] 1.465333e-02 -0.0038872126 0.0526407524 0.0412475653 0.0333480793
## [24,] 1.954358e-02 -0.0100302028 0.0572132270 0.0496034994 0.0493913216
## [25,] 1.495298e-02 -0.0142737667 0.0453391417 0.0447523524 0.0363183930
## [26,] 4.908647e-03 -0.0051034068 0.0496087928 0.0395018717 0.0360865973
## [27,] 3.704105e-03 0.0068874759 0.0720447858 -0.0024755876 0.0129930280
## [28,] 9.391763e-03 -0.0104148189 -0.0537091120 0.1098994418 0.0613607792
## [29,] 3.358917e-03 0.0099566978 0.0796134591 -0.0259503044 -0.0004122913
## [30,] -1.917001e-02 0.0157663620 -0.0385921907 0.0122655683 -0.0286546842
## [31,] -1.524097e-02 0.0131051233 -0.0115105902 0.1078438906 0.0001634335
## [32,] -1.465080e-02 0.0097740163 -0.0211475265 0.0138848702 0.0461764744
## [33,] -1.295229e-02 0.0215029134 0.0017360235 -0.0304481370 -0.0113240171
## [34,] -6.601193e-03 0.0288206682 0.0052309297 -0.0220584783 -0.0065160484
## [35,] -6.224033e-03 0.0289383738 0.0021149276 -0.0035059057 -0.0043587799
## [36,] -1.079981e-03 0.0187972793 0.0044882082 -0.0025086953 0.0021579337
## [37,] -4.485427e-03 0.0049005263 -0.0008620536 -0.0050592129 -0.0031700368
## [38,] -6.637146e-03 0.0215367064 0.0055728175 -0.0061989731 -0.0014844076
## [39,] 4.026890e-03 0.0180535934 0.0042851123 0.0004059684 -0.0018730783
## [40,] -2.458955e-02 -0.0035182081 0.0223313580 0.0050202770 0.0176300084
## [41,] 2.393814e-03 0.0011181282 0.0028572920 0.0101625571 0.0024071431
## [42,] -3.928372e-02 0.0042853457 -0.1881766846 -0.0129195896 -0.0074190766
## [43,] -6.105479e-02 -0.0271053744 0.0178245506 -0.1057697631 0.0350454501
## [44,] -4.937874e-02 -0.0172296043 -0.0066161324 0.0061481908 -0.1990578033
## [45,] -5.836645e-02 -0.0110431312 -0.0264523774 -0.0094468771 0.0268459632
## [46,] -4.653032e-02 -0.0176755177 -0.0108442432 -0.0119473136 0.0046056483
## [47,] -5.358833e-02 -0.0200814415 -0.0175144129 0.0055311480 -0.0219545646
## [48,] -5.162047e-02 -0.0258775896 -0.0100333134 -0.0024332523 -0.0037737394
## [49,] -3.559652e-02 -0.0211591601 -0.0076982738 -0.0004583106 -0.0039689096
## [50,] -2.825869e-02 -0.0340259211 0.0132704723 0.0182073970 0.0138555225
## [51,] -6.047327e-04 -0.0258182183 -0.0107949434 0.0171422706 0.0117313529
## [52,] -8.375854e-02 -0.0218082123 0.0484875770 -0.0059452985 -0.0037032315
## [53,] 2.049140e-02 -0.0104309434 -0.0138010942 0.0161595856 0.0126824673
## [54,] -1.020901e-02 -0.0374301127 -0.0211102545 0.0075658279 0.0196640620
## [55,] -1.795302e-02 -0.0457783057 0.0253386903 -0.0088998758 0.0168603364
## [56,] -1.491377e-02 -0.0287688242 0.0165779206 0.0220283358 -0.0315615880
## [57,] -1.680990e-02 -0.0495195041 0.0162950299 0.0130501084 0.0207189220
## [58,] -1.390938e-02 -0.0372089710 0.0099525890 0.0341609078 0.0195169423
## [59,] -2.495793e-02 -0.0501930591 0.0117159623 -0.0049922873 0.0013617384
## [60,] -2.232423e-02 -0.0379921136 0.0112608763 0.0106829226 0.0062238459
## [61,] -2.694170e-02 -0.0275582711 0.0015565738 0.0006178859 -0.0005272086
## [62,] -3.115618e-02 -0.0376627339 -0.0012103870 0.0017049453 -0.0010980369
## [63,] -2.384846e-02 -0.0224723303 -0.0086415036 -0.0023993601 -0.0000041520
## [64,] -5.142753e-02 -0.0852799873 0.0248867928 0.0185190816 0.0022506025
## [65,] -3.254522e-02 -0.0057662276 -0.0169423414 -0.0104708540 -0.0038647662
## [66,] 1.170030e-03 0.0007830403 -0.0033507079 0.0004893545 -0.0001856978
## [,21] [,22] [,23] [,24] [,25]
## [1,] 0.0009905912 1.124637e-03 -2.047138e-05 0.0026382395 0.0014254522
## [2,] -0.0172679139 -8.411361e-03 -7.761985e-03 -0.0102166776 -0.0095139101
## [3,] 0.0128424858 -3.443568e-03 2.467268e-03 -0.0035820355 -0.0019130625
## [4,] -0.0222915417 1.035762e-02 2.490237e-03 0.0062242137 0.0009606913
## [5,] 0.0247058699 -9.562092e-03 -1.262920e-04 -0.0108061411 -0.0040557053
## [6,] -0.0105548402 1.666236e-02 2.936495e-03 0.0053273644 -0.0053449889
## [7,] 0.0080357848 -9.152857e-03 -2.477981e-02 -0.0514099480 0.0033400592
## [8,] -0.0127841342 -5.199630e-04 -1.650246e-02 0.0730009407 -0.0114210684
## [9,] -0.0009330031 -4.624456e-03 5.210460e-03 -0.0152753918 0.0311337425
## [10,] -0.0078869284 -6.277831e-03 -1.053299e-03 -0.0125677258 -0.0125742491
## [11,] -0.0038931350 -2.826602e-03 -4.031060e-03 -0.0025448441 -0.0058487753
## [12,] 0.0080951712 4.869116e-03 1.208416e-02 0.0035136597 0.0031001372
## [13,] -0.0012260508 4.925614e-05 1.944954e-03 -0.0012330011 0.0064206111
## [14,] -0.0733342168 -6.465311e-02 -5.961749e-02 -0.0688709420 -0.0558239321
## [15,] 0.0077739136 4.932466e-03 -7.944696e-03 0.0051271832 0.0157941997
## [16,] 0.0215656378 1.584630e-02 1.465333e-02 0.0195435832 0.0149529833
## [17,] -0.0121960171 -1.124398e-02 -3.887213e-03 -0.0100302028 -0.0142737667
## [18,] 0.0602407700 5.602241e-02 5.264075e-02 0.0572132270 0.0453391417
## [19,] 0.0824856890 4.022872e-02 4.124757e-02 0.0496034994 0.0447523524
## [20,] -0.0646324792 6.954294e-02 3.334808e-02 0.0493913216 0.0363183930
## [21,] 0.3985634647 -7.230120e-02 8.053227e-02 0.0439775496 0.0515345259
## [22,] -0.0723012018 4.814463e-01 -8.584341e-02 0.0732963339 0.0361663814
## [23,] 0.0805322729 -8.584341e-02 4.941761e-01 -0.0582460239 0.0689971979
## [24,] 0.0439775496 7.329633e-02 -5.824602e-02 0.4048288548 -0.0618108665
## [25,] 0.0515345259 3.616638e-02 6.899720e-02 -0.0618108665 0.4876428254
## [26,] 0.0479597816 4.708737e-02 3.269186e-02 0.0597292754 -0.0732938569
## [27,] 0.0159775934 1.250564e-02 1.988506e-02 0.0091625149 0.0526224773
## [28,] 0.0871867164 7.778509e-02 7.688977e-02 0.0809293817 0.0796569739
## [29,] -0.0024460183 -2.387936e-03 -4.610441e-04 -0.0024943591 -0.0093618099
## [30,] 0.0016892519 -3.303991e-03 4.028920e-03 -0.0052882224 -0.0094146334
## [31,] -0.0101833530 -8.449178e-03 2.852316e-03 -0.0080118565 -0.0108513670
## [32,] -0.0222847109 -7.446789e-03 3.933102e-03 -0.0047236307 -0.0088418078
## [33,] 0.0103195083 1.435111e-02 1.362682e-02 -0.0069728316 -0.0106288516
## [34,] -0.0009929918 1.453538e-02 -1.947406e-02 0.0130949342 -0.0084622610
## [35,] 0.0042632502 -2.543793e-02 -2.686682e-02 -0.0039982020 -0.0052469569
## [36,] -0.0037023467 1.963643e-03 -2.724648e-02 0.0068838175 0.0086986000
## [37,] -0.0029486063 -5.136123e-04 -1.138956e-02 -0.0049065696 0.0757103779
## [38,] 0.0012264302 3.225537e-03 1.244339e-02 0.0017668467 -0.0029284061
## [39,] 0.0003825248 9.226432e-04 1.406026e-03 0.0045086459 -0.0056230989
## [40,] 0.0235028581 2.293748e-02 2.740843e-02 0.0217168675 0.0129583713
## [41,] 0.0031804225 1.904314e-03 2.249450e-03 0.0023584972 0.0082144734
## [42,] -0.0203083083 -1.237865e-02 -9.964098e-03 -0.0139905382 -0.0136767459
## [43,] -0.0224161773 -1.151182e-02 -1.288178e-02 -0.0167454048 -0.0103495015
## [44,] 0.0144628968 -6.016344e-03 6.602750e-03 -0.0032187651 -0.0069998486
## [45,] -0.1161079657 2.394880e-02 -2.986009e-02 -0.0198359819 -0.0146234040
## [46,] -0.0246733530 -2.075277e-01 5.568205e-02 0.0165340797 -0.0125798747
## [47,] 0.0299054286 4.182268e-02 -2.160156e-01 0.0041046787 -0.0096217584
## [48,] -0.0089346925 -8.080244e-03 2.013930e-03 -0.1933558282 0.0349063248
## [49,] -0.0014179692 -1.289954e-02 3.876961e-02 0.0272866921 -0.1724471170
## [50,] 0.0136589692 1.669820e-02 4.506235e-03 0.0170937021 0.0073643298
## [51,] 0.0128686433 1.056399e-02 1.133465e-02 0.0072908316 0.0412337934
## [52,] -0.0054387827 -9.103506e-04 1.046679e-03 -0.0059426914 0.0071490221
## [53,] 0.0164750373 1.398667e-02 1.228483e-02 0.0153814879 0.0066927125
## [54,] 0.0129779085 1.331876e-02 6.685595e-03 0.0131984076 0.0152347729
## [55,] 0.0091313902 1.123348e-02 5.313678e-03 0.0091110676 0.0120273031
## [56,] 0.0219066469 8.841311e-03 -2.892239e-03 0.0090327129 0.0104888489
## [57,] -0.0021539325 1.372027e-02 6.852178e-03 0.0111459749 0.0145347151
## [58,] -0.0065761205 -4.321186e-02 1.207730e-02 0.0191884403 0.0095007730
## [59,] 0.0093399196 6.564501e-02 3.978934e-02 -0.0006987365 0.0107786158
## [60,] 0.0095025959 2.585564e-03 3.146309e-02 0.0372453554 -0.0125694862
## [61,] -0.0021165105 1.478529e-03 -1.982739e-02 -0.0089135226 0.0305695904
## [62,] -0.0033454646 -2.519627e-03 -3.437524e-03 -0.0057299641 0.0120083138
## [63,] -0.0021546259 -1.026854e-03 -4.365973e-03 -0.0016010097 -0.0139470508
## [64,] 0.0007934607 3.639385e-03 -5.864836e-03 0.0006101185 -0.0033851188
## [65,] -0.0076409833 -3.111484e-03 -5.038822e-03 -0.0043061557 -0.0116242894
## [66,] -0.0020667251 -2.289379e-03 -3.596608e-03 0.0017223553 0.0024737492
## [,26] [,27] [,28] [,29] [,30]
## [1,] 0.0027671523 0.0115530410 -0.0111496329 0.0034937018 0.0045932640
## [2,] -0.0089495712 -0.0197241838 0.0229455662 -0.0153776881 -0.1619183584
## [3,] -0.0002779557 -0.0013991342 -0.0043476084 0.0094426847 0.0367043535
## [4,] 0.0013535450 -0.0095705757 0.0241059290 -0.0058834187 -0.0092290957
## [5,] -0.0015901109 -0.0089008510 0.0091313209 0.0003000915 0.0090283630
## [6,] -0.0027140856 -0.0081739379 0.0101569521 -0.0021555264 0.0029186075
## [7,] 0.0083296722 -0.0090877745 0.0220711068 0.0026794133 0.0069442604
## [8,] -0.0087221226 -0.0032847085 0.0052140419 -0.0047344491 0.0049704428
## [9,] -0.0193556733 -0.0162965625 0.0134730938 0.0085630785 0.0001276787
## [10,] 0.0987053449 -0.0088092767 -0.0094599783 -0.0280914594 0.0063957015
## [11,] -0.0340052573 -0.0309799420 0.0321253121 0.0374506698 -0.0053939864
## [12,] -0.0079875021 0.0189866770 -0.0030250635 -0.0252042353 0.0155527991
## [13,] -0.0029626717 0.0361329141 -0.0218646687 -0.1108706366 0.0012944623
## [14,] -0.0556029090 -0.0184613091 -0.1007316314 0.0030964330 0.0055627539
## [15,] -0.0122762870 -0.0195820650 -0.0206003098 -0.0155605885 -0.0608850550
## [16,] 0.0049086466 0.0037041054 0.0093917628 0.0033589174 -0.0191700091
## [17,] -0.0051034068 0.0068874759 -0.0104148189 0.0099566978 0.0157663620
## [18,] 0.0496087928 0.0720447858 -0.0537091120 0.0796134591 -0.0385921907
## [19,] 0.0395018717 -0.0024755876 0.1098994418 -0.0259503044 0.0122655683
## [20,] 0.0360865973 0.0129930280 0.0613607792 -0.0004122913 -0.0286546842
## [21,] 0.0479597816 0.0159775934 0.0871867164 -0.0024460183 0.0016892519
## [22,] 0.0470873668 0.0125056375 0.0777850880 -0.0023879361 -0.0033039911
## [23,] 0.0326918648 0.0198850648 0.0768897663 -0.0004610441 0.0040289204
## [24,] 0.0597292754 0.0091625149 0.0809293817 -0.0024943591 -0.0052882224
## [25,] -0.0732938569 0.0526224773 0.0796569739 -0.0093618099 -0.0094146334
## [26,] 0.5980064558 -0.1487498475 0.0124006753 0.0102321875 0.0112843160
## [27,] -0.1487498475 0.5382050953 0.0350987734 -0.0517660899 -0.0044689883
## [28,] 0.0124006753 0.0350987734 0.6082627308 0.0505171977 0.0420194116
## [29,] 0.0102321875 -0.0517660899 0.0505171977 0.5641712011 -0.0085168668
## [30,] 0.0112843160 -0.0044689883 0.0420194116 -0.0085168668 0.6906966151
## [31,] -0.0017490551 0.0045191935 0.0031585707 0.0036598110 -0.0414641099
## [32,] 0.0051155087 0.0008423422 0.0122662212 0.0009759210 0.0355431223
## [33,] 0.0060365596 0.0096919001 0.0103025519 0.0088824646 0.0209051733
## [34,] 0.0082587029 0.0130348191 0.0160754834 0.0089668389 0.0311246598
## [35,] 0.0104000167 0.0139339890 0.0235674636 0.0097594118 0.0396415958
## [36,] 0.0022371971 0.0117897750 0.0195348433 0.0049536192 0.0270032793
## [37,] -0.0002275872 0.0014975032 0.0062569335 0.0082401365 0.0124479634
## [38,] 0.0449843932 -0.0358092927 -0.0033908968 -0.0004046461 0.0345794980
## [39,] -0.0388373216 -0.0911692265 0.0075892470 0.0479816929 0.0453716606
## [40,] 0.0074698441 0.0033291503 -0.0109491633 -0.0517410239 -0.0319446867
## [41,] -0.0116172940 0.0281644702 -0.0588823014 -0.1644537928 0.0360479393
## [42,] -0.0040655100 -0.0145387940 0.0391825973 -0.0243600305 -0.1611309530
## [43,] -0.0023071203 -0.0025461081 -0.0149106121 -0.0006269794 0.0723154405
## [44,] 0.0028607637 0.0008566042 0.0025096656 -0.0017683566 0.0060364863
## [45,] -0.0023139799 -0.0020187846 -0.0031361669 -0.0030200354 0.0328497116
## [46,] 0.0022903616 -0.0021515663 0.0025703770 -0.0046692456 0.0094758323
## [47,] 0.0002723998 -0.0010527307 0.0041816978 -0.0038088073 0.0228730092
## [48,] 0.0245697281 -0.0160021861 -0.0010638109 -0.0061548396 0.0116465968
## [49,] -0.0331378364 0.0278011417 0.0208304321 0.0061745147 0.0054855285
## [50,] -0.2028860386 -0.0780759110 0.0041674763 0.0092417866 -0.0026143970
## [51,] -0.0639551113 -0.1954143580 0.0284954966 0.0287174174 -0.0101121493
## [52,] -0.0070594573 -0.0030352531 -0.1437869877 0.0183116814 0.0297959804
## [53,] 0.0195503102 0.0028102983 0.0298814881 -0.1587028970 -0.0031868674
## [54,] 0.0069622195 -0.0008744469 0.0104048152 -0.0053935949 -0.1639790787
## [55,] 0.0067169958 -0.0044089100 0.0118246155 -0.0075879118 -0.0030354815
## [56,] 0.0103639016 0.0018062115 0.0180972314 -0.0032963732 0.0136204137
## [57,] 0.0054394425 -0.0055588092 0.0111261407 -0.0092675959 -0.0169333533
## [58,] 0.0107573567 -0.0035114542 0.0196117530 -0.0077917616 0.0102692145
## [59,] -0.0014292657 -0.0055743006 -0.0003622190 -0.0073857574 -0.0225625949
## [60,] 0.0111757557 -0.0118011128 0.0036729331 -0.0130435515 0.0007983492
## [61,] -0.0130703166 0.0293780376 0.0284116767 0.0161001550 0.0063774196
## [62,] 0.0318301289 0.0575276360 0.0157411733 -0.0217201417 0.0001260029
## [63,] 0.0582830426 0.0515255763 0.0037408808 -0.0203009885 -0.0162730283
## [64,] 0.0384262139 -0.0030165825 0.0263094176 -0.0169106353 0.0355395545
## [65,] 0.0251594869 -0.0111199347 -0.0001743791 0.0243429041 -0.0082050518
## [66,] -0.0062964122 0.0022259379 -0.0006461750 -0.0020588684 -0.0041371575
## [,31] [,32] [,33] [,34] [,35]
## [1,] 0.0044852495 0.0029378567 1.044540e-02 0.0116465260 0.0096331652
## [2,] 0.0230946573 0.0160789042 -2.613912e-04 -0.0026999051 -0.0018545111
## [3,] -0.1014358021 -0.0048570093 -1.329383e-02 -0.0060279160 0.0016260669
## [4,] -0.0140850045 -0.1012572859 3.704902e-02 -0.0026804089 -0.0112856736
## [5,] 0.0067107654 0.0035761798 -1.437520e-01 0.0177468264 0.0106749561
## [6,] -0.0037197323 0.0042803388 2.152799e-02 -0.1384300581 0.0024036094
## [7,] -0.0036345263 -0.0013275269 2.371236e-03 -0.0071932826 -0.1595215824
## [8,] 0.0060218490 0.0058419516 -3.956808e-03 0.0196557429 0.0311265614
## [9,] -0.0012705412 -0.0004914935 -1.991595e-03 -0.0135815109 -0.0093338021
## [10,] 0.0052388762 0.0048047717 8.126642e-04 0.0074335223 0.0053277888
## [11,] -0.0029944965 0.0010476937 -8.593450e-03 -0.0132798654 -0.0094172069
## [12,] 0.0027180780 -0.0011020247 6.152593e-03 0.0085786648 0.0116581279
## [13,] 0.0062799729 0.0073047397 3.132205e-03 0.0051154079 0.0053315483
## [14,] 0.0128118910 0.0074101521 7.294553e-03 0.0023920351 -0.0006092628
## [15,] -0.0477656194 -0.0402089258 -6.783888e-02 -0.0788821200 -0.0774130187
## [16,] -0.0152409724 -0.0146508038 -1.295229e-02 -0.0066011932 -0.0062240326
## [17,] 0.0131051233 0.0097740163 2.150291e-02 0.0288206682 0.0289383738
## [18,] -0.0115105902 -0.0211475265 1.736023e-03 0.0052309297 0.0021149276
## [19,] 0.1078438906 0.0138848702 -3.044814e-02 -0.0220584783 -0.0035059057
## [20,] 0.0001634335 0.0461764744 -1.132402e-02 -0.0065160484 -0.0043587799
## [21,] -0.0101833530 -0.0222847109 1.031951e-02 -0.0009929918 0.0042632502
## [22,] -0.0084491782 -0.0074467892 1.435111e-02 0.0145353815 -0.0254379256
## [23,] 0.0028523165 0.0039331016 1.362682e-02 -0.0194740584 -0.0268668207
## [24,] -0.0080118565 -0.0047236307 -6.972832e-03 0.0130949342 -0.0039982020
## [25,] -0.0108513670 -0.0088418078 -1.062885e-02 -0.0084622610 -0.0052469569
## [26,] -0.0017490551 0.0051155087 6.036560e-03 0.0082587029 0.0104000167
## [27,] 0.0045191935 0.0008423422 9.691900e-03 0.0130348191 0.0139339890
## [28,] 0.0031585707 0.0122662212 1.030255e-02 0.0160754834 0.0235674636
## [29,] 0.0036598110 0.0009759210 8.882465e-03 0.0089668389 0.0097594118
## [30,] -0.0414641099 0.0355431223 2.090517e-02 0.0311246598 0.0396415958
## [31,] 0.7066082287 -0.0816284315 6.195127e-02 0.0495229334 0.0268662957
## [32,] -0.0816284315 0.7411256240 -2.839520e-02 0.0479609582 0.0190539895
## [33,] 0.0619512658 -0.0283951959 5.885036e-01 -0.1015380994 0.0761686037
## [34,] 0.0495229334 0.0479609582 -1.015381e-01 0.6023047719 -0.0929272157
## [35,] 0.0268662957 0.0190539895 7.616860e-02 -0.0929272157 0.5725782740
## [36,] 0.0167808631 0.0162055873 2.034655e-02 0.0402628474 -0.0596552287
## [37,] 0.0109083227 0.0085914116 1.651560e-02 0.0105088851 0.0248368621
## [38,] 0.0251231317 0.0225759274 3.515803e-02 0.0454025980 0.0348757503
## [39,] 0.0139750633 0.0089584441 2.017755e-02 0.0223044466 0.0230225770
## [40,] 0.0252046636 0.0285059375 3.782276e-02 0.0415819264 0.0444258622
## [41,] -0.0037277015 -0.0058301582 -6.471190e-03 -0.0087247491 -0.0060194242
## [42,] -0.0333539952 0.0175480637 2.787745e-02 0.0255081700 0.0219885936
## [43,] 0.0574855530 0.0387726829 3.999650e-03 -0.0088778925 -0.0044329648
## [44,] 0.0397257769 -0.1000288108 6.025124e-02 0.0582121923 -0.0045909862
## [45,] -0.0002090901 0.0371083318 -5.414617e-02 -0.0091269477 0.0164780310
## [46,] 0.0134744423 0.0146170839 1.517312e-02 -0.0044486096 0.0221294377
## [47,] -0.0027339978 0.0059866825 -3.066608e-02 0.0433685783 0.0337413271
## [48,] 0.0054142827 0.0087015150 2.841669e-03 -0.0119017732 -0.0037170661
## [49,] 0.0032139623 0.0040144802 -1.301667e-03 -0.0042483479 -0.0150228002
## [50,] -0.0057107139 -0.0015653016 -1.067413e-02 -0.0140424660 -0.0188413031
## [51,] -0.0108468271 -0.0054213620 -9.145045e-03 -0.0099459320 -0.0113955434
## [52,] 0.0285635489 0.0179415407 1.355674e-02 0.0052048370 0.0066785728
## [53,] -0.0082865445 -0.0008501486 1.811041e-03 0.0036598405 0.0041947514
## [54,] -0.0013752373 -0.0151558549 -1.561341e-02 -0.0204977801 -0.0211956602
## [55,] -0.1724997665 0.0391262767 -7.190498e-03 -0.0177993173 -0.0202134958
## [56,] -0.0040417009 -0.2427388081 -5.343361e-02 -0.0369788535 0.0096030609
## [57,] -0.0019874327 0.0204718055 -1.302140e-01 0.0344121640 -0.0426980726
## [58,] -0.0451495758 -0.0180596044 -3.157180e-02 -0.1937802574 0.0435557977
## [59,] 0.0227605664 0.0027099528 3.468144e-02 0.0158351936 -0.2003819682
## [60,] -0.0010815190 0.0007178985 -4.128837e-03 -0.0092962666 0.0162174918
## [61,] 0.0074815475 0.0057906434 7.481058e-03 -0.0041560291 0.0176574177
## [62,] 0.0012658243 0.0014278514 -3.785176e-03 -0.0105085502 -0.0102769940
## [63,] -0.0067502962 -0.0006726951 -6.847849e-03 -0.0129388328 -0.0094127367
## [64,] 0.0114441878 0.0035852475 -6.747377e-03 -0.0181531198 -0.0229138158
## [65,] 0.0023271649 0.0084658492 6.995576e-03 0.0074794065 0.0018988666
## [66,] 0.0012970365 -0.0026370886 -8.338188e-05 0.0002255103 0.0007055726
## [,36] [,37] [,38] [,39] [,40]
## [1,] 0.0082344306 0.0010932290 0.0108259251 1.431648e-02 -0.008427393
## [2,] -0.0012174804 0.0009841793 -0.0031613652 -6.569297e-03 0.003743339
## [3,] -0.0031510469 0.0015899668 -0.0038847634 -1.255757e-02 0.020907935
## [4,] -0.0004632151 -0.0004684053 -0.0037917941 -8.660879e-03 0.011078380
## [5,] -0.0040580096 0.0029220415 -0.0033756462 -9.665108e-03 0.015355902
## [6,] 0.0031159629 -0.0025220752 0.0010058367 -8.899555e-03 0.014143194
## [7,] 0.0219310514 0.0180215271 -0.0024535370 -1.557394e-03 0.019727688
## [8,] -0.1574426795 -0.0040405982 -0.0036658425 -1.536044e-02 0.012258405
## [9,] 0.0170328667 -0.1203510718 0.0135640518 7.408325e-03 0.019711639
## [10,] 0.0001583417 0.0137242344 -0.1169015851 -2.402603e-02 0.001293312
## [11,] -0.0078602878 0.0000804220 -0.0072144930 -1.172719e-01 0.036026140
## [12,] 0.0053795837 0.0056449669 0.0085850645 2.712837e-02 -0.112521214
## [13,] 0.0044086497 0.0006421238 0.0091060211 2.998261e-02 -0.005914378
## [14,] -0.0036522306 0.0035082619 -0.0019547449 -1.747055e-03 -0.024746394
## [15,] -0.0548535820 -0.0228266312 -0.0672227179 -3.791048e-02 -0.061596551
## [16,] -0.0010799814 -0.0044854273 -0.0066371458 4.026890e-03 -0.024589548
## [17,] 0.0187972793 0.0049005263 0.0215367064 1.805359e-02 -0.003518208
## [18,] 0.0044882082 -0.0008620536 0.0055728175 4.285112e-03 0.022331358
## [19,] -0.0025086953 -0.0050592129 -0.0061989731 4.059684e-04 0.005020277
## [20,] 0.0021579337 -0.0031700368 -0.0014844076 -1.873078e-03 0.017630008
## [21,] -0.0037023467 -0.0029486063 0.0012264302 3.825248e-04 0.023502858
## [22,] 0.0019636429 -0.0005136123 0.0032255369 9.226432e-04 0.022937482
## [23,] -0.0272464795 -0.0113895614 0.0124433909 1.406026e-03 0.027408433
## [24,] 0.0068838175 -0.0049065696 0.0017668467 4.508646e-03 0.021716868
## [25,] 0.0086986000 0.0757103779 -0.0029284061 -5.623099e-03 0.012958371
## [26,] 0.0022371971 -0.0002275872 0.0449843932 -3.883732e-02 0.007469844
## [27,] 0.0117897750 0.0014975032 -0.0358092927 -9.116923e-02 0.003329150
## [28,] 0.0195348433 0.0062569335 -0.0033908968 7.589247e-03 -0.010949163
## [29,] 0.0049536192 0.0082401365 -0.0004046461 4.798169e-02 -0.051741024
## [30,] 0.0270032793 0.0124479634 0.0345794980 4.537166e-02 -0.031944687
## [31,] 0.0167808631 0.0109083227 0.0251231317 1.397506e-02 0.025204664
## [32,] 0.0162055873 0.0085914116 0.0225759274 8.958444e-03 0.028505938
## [33,] 0.0203465452 0.0165156045 0.0351580332 2.017755e-02 0.037822756
## [34,] 0.0402628474 0.0105088851 0.0454025980 2.230445e-02 0.041581926
## [35,] -0.0596552287 0.0248368621 0.0348757503 2.302258e-02 0.044425862
## [36,] 0.6181074143 -0.0348577212 0.0341873646 1.622260e-02 0.031738009
## [37,] -0.0348577212 0.7835551887 -0.0503542213 1.772782e-02 0.023497426
## [38,] 0.0341873646 -0.0503542213 0.7069738731 -1.642897e-01 -0.025376765
## [39,] 0.0162226050 0.0177278243 -0.1642897202 6.875261e-01 -0.002922829
## [40,] 0.0317380087 0.0234974257 -0.0253767655 -2.922829e-03 0.815610078
## [41,] -0.0062771984 0.0008766779 0.0093039058 4.370918e-03 0.031578201
## [42,] 0.0137414783 0.0076610747 0.0191867878 3.724131e-03 0.035165454
## [43,] -0.0069243786 0.0013412928 -0.0023982167 -1.041398e-02 0.020414828
## [44,] -0.0043478259 0.0050785208 0.0061067638 -2.619909e-03 0.025987982
## [45,] 0.0110544997 0.0055289013 0.0139532744 -1.282833e-04 0.030327593
## [46,] 0.0209728297 0.0016599873 0.0030050468 -4.218342e-03 0.020372994
## [47,] -0.0064659771 0.0078639925 0.0010369956 -4.223460e-03 0.026530687
## [48,] 0.0853664887 0.0294693894 -0.0043965955 -5.787793e-03 0.020784777
## [49,] 0.0251294321 0.0582375776 -0.0410975021 -1.487081e-02 0.022175879
## [50,] -0.0124879055 0.0110590561 0.0365055863 5.913616e-02 0.035949314
## [51,] -0.0030461208 -0.0122153154 0.0329522041 -2.356010e-02 0.009704547
## [52,] -0.0008406137 -0.0003688363 0.0205736754 -1.472481e-02 -0.003823182
## [53,] 0.0056692349 0.0070275229 -0.0002721699 4.833602e-05 -0.062332301
## [54,] -0.0134369816 -0.0042793422 -0.0160149979 -2.871770e-02 0.042041252
## [55,] -0.0123020375 -0.0018783211 -0.0128849221 -1.116027e-02 0.006323155
## [56,] 0.0092025187 0.0015176594 0.0034582847 -3.437152e-03 0.023088574
## [57,] -0.0194340680 -0.0043876149 -0.0191290900 -1.652244e-02 0.005078087
## [58,] 0.0110094230 -0.0012328970 -0.0072104379 -9.111820e-03 0.014349467
## [59,] 0.0004513667 -0.0022332963 -0.0219185454 -1.719229e-02 0.005505614
## [60,] -0.2433983317 -0.0324212208 0.0426194395 -9.412533e-03 0.011206521
## [61,] -0.0089762188 -0.1877971855 -0.1145288190 -6.328730e-03 0.037068878
## [62,] -0.0058876473 -0.0021033168 -0.1705208566 4.878245e-03 0.016027251
## [63,] -0.0111654676 0.0085965603 0.0154143475 -1.457908e-01 0.045573525
## [64,] -0.0121429037 -0.0010731077 0.0059237874 -7.912885e-03 -0.190474279
## [65,] 0.0030129565 -0.0125180928 0.0168483703 -7.011704e-02 0.049606744
## [66,] 0.0014262393 -0.0005401815 -0.0016109073 1.325791e-03 -0.003238743
## [,41] [,42] [,43] [,44] [,45]
## [1,] -0.0071562177 0.0049023061 -0.0100889045 -0.0023998360 -0.0007663466
## [2,] 0.0068308752 -0.0037615938 0.0014170970 -0.0064290921 0.0078585955
## [3,] 0.0026213862 0.0069189911 -0.0060749219 0.0126928159 -0.0002287436
## [4,] 0.0063393912 -0.0018881997 0.0061909356 0.0184626655 0.0014044917
## [5,] 0.0053641775 -0.0025439472 0.0058901702 -0.0027625487 0.0090972442
## [6,] 0.0041506954 0.0023587695 0.0107924819 0.0029372379 0.0088585513
## [7,] 0.0054434431 -0.0038182673 0.0031180789 -0.0138412293 0.0056422426
## [8,] 0.0049529477 0.0048398081 0.0112551181 0.0108639823 0.0061588453
## [9,] 0.0070002297 -0.0023487351 0.0091369153 0.0016206451 0.0026977396
## [10,] -0.0083788455 0.0045658150 0.0100107699 0.0057058802 0.0074822064
## [11,] 0.0256608120 -0.0138128212 0.0084578950 -0.0018403275 0.0009136342
## [12,] -0.0045015574 0.0225132401 0.0065535636 0.0107678216 0.0021290962
## [13,] -0.1973703212 -0.0084972636 0.0032961168 -0.0011685685 0.0045865335
## [14,] -0.0036784659 0.0229421640 0.0246423003 0.0148523490 0.0278851731
## [15,] 0.0131160500 -0.0377273136 0.0073635222 -0.0103747110 -0.0240587188
## [16,] 0.0023938143 -0.0392837206 -0.0610547875 -0.0493787399 -0.0583664498
## [17,] 0.0011181282 0.0042853457 -0.0271053744 -0.0172296043 -0.0110431312
## [18,] 0.0028572920 -0.1881766846 0.0178245506 -0.0066161324 -0.0264523774
## [19,] 0.0101625571 -0.0129195896 -0.1057697631 0.0061481908 -0.0094468771
## [20,] 0.0024071431 -0.0074190766 0.0350454501 -0.1990578033 0.0268459632
## [21,] 0.0031804225 -0.0203083083 -0.0224161773 0.0144628968 -0.1161079657
## [22,] 0.0019043145 -0.0123786542 -0.0115118234 -0.0060163439 0.0239487975
## [23,] 0.0022494496 -0.0099640980 -0.0128817827 0.0066027496 -0.0298600865
## [24,] 0.0023584972 -0.0139905382 -0.0167454048 -0.0032187651 -0.0198359819
## [25,] 0.0082144734 -0.0136767459 -0.0103495015 -0.0069998486 -0.0146234040
## [26,] -0.0116172940 -0.0040655100 -0.0023071203 0.0028607637 -0.0023139799
## [27,] 0.0281644702 -0.0145387940 -0.0025461081 0.0008566042 -0.0020187846
## [28,] -0.0588823014 0.0391825973 -0.0149106121 0.0025096656 -0.0031361669
## [29,] -0.1644537928 -0.0243600305 -0.0006269794 -0.0017683566 -0.0030200354
## [30,] 0.0360479393 -0.1611309530 0.0723154405 0.0060364863 0.0328497116
## [31,] -0.0037277015 -0.0333539952 0.0574855530 0.0397257769 -0.0002090901
## [32,] -0.0058301582 0.0175480637 0.0387726829 -0.1000288108 0.0371083318
## [33,] -0.0064711896 0.0278774517 0.0039996504 0.0602512400 -0.0541461749
## [34,] -0.0087247491 0.0255081700 -0.0088778925 0.0582121923 -0.0091269477
## [35,] -0.0060194242 0.0219885936 -0.0044329648 -0.0045909862 0.0164780310
## [36,] -0.0062771984 0.0137414783 -0.0069243786 -0.0043478259 0.0110544997
## [37,] 0.0008766779 0.0076610747 0.0013412928 0.0050785208 0.0055289013
## [38,] 0.0093039058 0.0191867878 -0.0023982167 0.0061067638 0.0139532744
## [39,] 0.0043709179 0.0037241314 -0.0104139819 -0.0026199086 -0.0001282833
## [40,] 0.0315782011 0.0351654540 0.0204148280 0.0259879821 0.0303275934
## [41,] 0.7906328829 -0.0090991424 -0.0004663476 -0.0015685422 -0.0036983535
## [42,] -0.0090991424 0.2611470151 -0.0237345391 0.0431536339 0.0376301583
## [43,] -0.0004663476 -0.0237345391 0.1273715731 0.0141829098 0.0690300156
## [44,] -0.0015685422 0.0431536339 0.0141829098 0.2504227096 -0.0298783911
## [45,] -0.0036983535 0.0376301583 0.0690300156 -0.0298783911 0.1381497527
## [46,] -0.0010233711 0.0315436576 0.0453536110 0.0659609656 0.0114088058
## [47,] -0.0009052001 0.0321241759 0.0604323940 0.0051091695 0.0844862629
## [48,] -0.0047279510 0.0286513893 0.0573958000 0.0434169573 0.0497803782
## [49,] 0.0201428527 0.0197012819 0.0398304194 0.0285666312 0.0349405049
## [50,] -0.0135538082 0.0089525724 0.0364335660 0.0270941829 0.0252367566
## [51,] 0.0172726808 0.0171765953 -0.0009702544 0.0034730586 -0.0020339755
## [52,] -0.0658950641 -0.0008437656 0.1029914284 0.0720679525 0.0841609132
## [53,] -0.1289854441 0.0179250613 -0.0290966706 -0.0143060475 -0.0206283323
## [54,] -0.0235750772 0.0907859378 -0.0040740589 0.0148474894 0.0043890694
## [55,] -0.0002980855 0.0058543673 0.0112926244 0.0113763222 0.0142109303
## [56,] -0.0061108714 0.0057622944 0.0026763482 0.0210058737 0.0370839900
## [57,] -0.0013913243 -0.0032547701 0.0277091065 0.0270824739 0.0177076040
## [58,] -0.0041414443 -0.0036338212 0.0224847733 -0.0633622109 0.0565608937
## [59,] -0.0008269600 0.0055246973 0.0319829395 0.0824271089 -0.0157446245
## [60,] -0.0151791441 0.0069057207 0.0275921202 0.0199045356 0.0203229074
## [61,] 0.0264467467 0.0151169449 0.0290762396 0.0267648064 0.0239942994
## [62,] -0.0111401736 0.0131687156 0.0363387032 0.0277186199 0.0277813897
## [63,] -0.0189214878 0.0197069877 0.0273676951 0.0212877543 0.0208860300
## [64,] 0.0438253834 -0.0088995917 0.0656016927 0.0461701105 0.0449483808
## [65,] -0.1553056334 0.0381931443 0.0311714639 0.0282118198 0.0332246443
## [66,] 0.0008584699 0.0015101964 -0.0022162731 -0.0004286797 -0.0014353410
## [,46] [,47] [,48] [,49] [,50]
## [1,] -0.005395781 -0.0020300695 -0.0092344195 -0.0035202334 -0.006783795
## [2,] 0.005003123 0.0017629535 0.0052732048 0.0010156129 -0.001460957
## [3,] 0.002905069 0.0060188433 0.0090208998 0.0032129542 0.004794528
## [4,] 0.007664331 -0.0067142608 0.0032819502 -0.0004242595 0.002922520
## [5,] -0.004236789 0.0142903740 0.0098011625 0.0031102082 0.002833388
## [6,] -0.016902176 0.0080231452 0.0041298556 0.0032938360 0.002470836
## [7,] 0.013764859 0.0143629746 0.0225009050 -0.0016336666 0.001563866
## [8,] 0.011164740 0.0016509927 -0.0201310705 -0.0186461991 -0.001696989
## [9,] 0.003030303 0.0049870597 -0.0100192511 0.0272996454 0.004229534
## [10,] 0.007340643 0.0029090000 0.0162104859 -0.0149281682 0.003224213
## [11,] 0.002016070 0.0022566350 0.0042759070 0.0104506381 0.006626116
## [12,] 0.008019648 0.0021293934 0.0089601509 0.0068501448 -0.003787253
## [13,] 0.001777476 0.0021385677 -0.0006988548 0.0040177734 -0.014230742
## [14,] 0.014615875 0.0180893930 0.0136571212 0.0098287613 -0.013957204
## [15,] -0.002747691 -0.0072691130 0.0073141001 0.0078064506 0.026834764
## [16,] -0.046530323 -0.0535883254 -0.0516204716 -0.0355965208 -0.028258693
## [17,] -0.017675518 -0.0200814415 -0.0258775896 -0.0211591601 -0.034025921
## [18,] -0.010844243 -0.0175144129 -0.0100333134 -0.0076982738 0.013270472
## [19,] -0.011947314 0.0055311480 -0.0024332523 -0.0004583106 0.018207397
## [20,] 0.004605648 -0.0219545646 -0.0037737394 -0.0039689096 0.013855522
## [21,] -0.024673353 0.0299054286 -0.0089346925 -0.0014179692 0.013658969
## [22,] -0.207527680 0.0418226768 -0.0080802439 -0.0128995410 0.016698204
## [23,] 0.055682050 -0.2160156494 0.0020139298 0.0387696122 0.004506235
## [24,] 0.016534080 0.0041046787 -0.1933558282 0.0272866921 0.017093702
## [25,] -0.012579875 -0.0096217584 0.0349063248 -0.1724471170 0.007364330
## [26,] 0.002290362 0.0002723998 0.0245697281 -0.0331378364 -0.202886039
## [27,] -0.002151566 -0.0010527307 -0.0160021861 0.0278011417 -0.078075911
## [28,] 0.002570377 0.0041816978 -0.0010638109 0.0208304321 0.004167476
## [29,] -0.004669246 -0.0038088073 -0.0061548396 0.0061745147 0.009241787
## [30,] 0.009475832 0.0228730092 0.0116465968 0.0054855285 -0.002614397
## [31,] 0.013474442 -0.0027339978 0.0054142827 0.0032139623 -0.005710714
## [32,] 0.014617084 0.0059866825 0.0087015150 0.0040144802 -0.001565302
## [33,] 0.015173116 -0.0306660769 0.0028416693 -0.0013016668 -0.010674130
## [34,] -0.004448610 0.0433685783 -0.0119017732 -0.0042483479 -0.014042466
## [35,] 0.022129438 0.0337413271 -0.0037170661 -0.0150228002 -0.018841303
## [36,] 0.020972830 -0.0064659771 0.0853664887 0.0251294321 -0.012487906
## [37,] 0.001659987 0.0078639925 0.0294693894 0.0582375776 0.011059056
## [38,] 0.003005047 0.0010369956 -0.0043965955 -0.0410975021 0.036505586
## [39,] -0.004218342 -0.0042234600 -0.0057877928 -0.0148708104 0.059136159
## [40,] 0.020372994 0.0265306873 0.0207847767 0.0221758788 0.035949314
## [41,] -0.001023371 -0.0009052001 -0.0047279510 0.0201428527 -0.013553808
## [42,] 0.031543658 0.0321241759 0.0286513893 0.0197012819 0.008952572
## [43,] 0.045353611 0.0604323940 0.0573958000 0.0398304194 0.036433566
## [44,] 0.065960966 0.0051091695 0.0434169573 0.0285666312 0.027094183
## [45,] 0.011408806 0.0844862629 0.0497803782 0.0349405049 0.025236757
## [46,] 0.358258493 -0.1496445960 0.0528285001 0.0225428909 0.028598827
## [47,] -0.149644596 0.3192144638 0.0257923045 0.0430255613 0.026417370
## [48,] 0.052828500 0.0257923045 0.3169419000 -0.1718348055 0.054321955
## [49,] 0.022542891 0.0430255613 -0.1718348055 0.6763714715 -0.057233394
## [50,] 0.028598827 0.0264173695 0.0543219545 -0.0572333936 0.654395308
## [51,] 0.003044136 0.0030402548 -0.0015644290 0.0159402421 -0.100479761
## [52,] 0.069205664 0.0798331810 0.0749352046 0.0595333253 0.012660485
## [53,] -0.015708172 -0.0166866948 -0.0160439864 -0.0046979260 0.016498352
## [54,] 0.012789954 0.0123522242 0.0171596967 0.0146182158 0.025322152
## [55,] 0.015965881 0.0202348331 0.0243543786 0.0198562595 0.030457186
## [56,] 0.011620149 0.0410364198 0.0187667384 0.0153271583 0.020498770
## [57,] 0.005799196 0.0228537375 0.0234978511 0.0205642873 0.032712394
## [58,] 0.137725070 -0.0414742181 0.0274628856 0.0133423606 0.026515569
## [59,] -0.067450580 0.0084573356 0.0309644345 0.0287072057 0.030702540
## [60,] 0.017520064 0.0238863440 -0.1034137680 0.0660061992 0.054924612
## [61,] 0.023610218 0.0315945895 0.0900502351 -0.1386471692 -0.069641352
## [62,] 0.025944166 0.0315356373 0.0261503556 0.0436513486 -0.243861444
## [63,] 0.019998992 0.0247254570 0.0251750874 0.0375424668 0.062863817
## [64,] 0.044093234 0.0513895031 0.0607927629 0.0250500545 0.074574623
## [65,] 0.027359200 0.0271851336 0.0371376859 -0.0280714496 0.017739256
## [66,] 0.001003138 -0.0011074031 -0.0033948821 -0.0001775879 -0.001943416
## [,51] [,52] [,53] [,54] [,55]
## [1,] -2.364585e-04 -0.0027671840 3.821541e-03 0.0042945537 0.0064417892
## [2,] -3.457361e-03 -0.0029089369 -3.945616e-03 -0.0025510122 -0.0060657289
## [3,] -2.922482e-03 0.0084403319 -5.204399e-03 -0.0050286015 0.0049224436
## [4,] 5.355254e-06 -0.0037848615 -3.496736e-04 -0.0081622644 -0.0017888298
## [5,] -3.653583e-03 0.0055735956 -4.662755e-03 -0.0087720505 -0.0181082718
## [6,] -2.585977e-03 0.0045710181 -4.173075e-03 -0.0096883385 -0.0075183641
## [7,] -5.435892e-03 -0.0009245620 4.473532e-04 -0.0088444920 -0.0122142541
## [8,] -1.437877e-03 0.0084097174 -7.491182e-03 -0.0114954856 -0.0127563604
## [9,] -2.854375e-03 0.0038813864 -1.297237e-03 -0.0066476782 -0.0083237355
## [10,] -2.317383e-02 -0.0031713167 -9.753156e-03 -0.0115796947 -0.0131201275
## [11,] 1.789501e-03 0.0065478130 1.664096e-02 -0.0060094586 -0.0017450861
## [12,] -1.090782e-03 0.0083581237 -2.700902e-02 -0.0103528820 -0.0263253461
## [13,] 1.484821e-02 -0.0207728677 -4.007133e-02 -0.0075557934 -0.0009902677
## [14,] -1.444535e-02 0.0137335752 -2.039888e-02 -0.0146406521 -0.0102226959
## [15,] 2.056410e-02 -0.0112652160 -2.484728e-03 0.0357976537 0.0324782594
## [16,] -6.047327e-04 -0.0837585447 2.049140e-02 -0.0102090115 -0.0179530211
## [17,] -2.581822e-02 -0.0218082123 -1.043094e-02 -0.0374301127 -0.0457783057
## [18,] -1.079494e-02 0.0484875770 -1.380109e-02 -0.0211102545 0.0253386903
## [19,] 1.714227e-02 -0.0059452985 1.615959e-02 0.0075658279 -0.0088998758
## [20,] 1.173135e-02 -0.0037032315 1.268247e-02 0.0196640620 0.0168603364
## [21,] 1.286864e-02 -0.0054387827 1.647504e-02 0.0129779085 0.0091313902
## [22,] 1.056399e-02 -0.0009103506 1.398667e-02 0.0133187627 0.0112334813
## [23,] 1.133465e-02 0.0010466791 1.228483e-02 0.0066855946 0.0053136782
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## [63,] -1.734832e-03
## [64,] -2.039201e-03
## [65,] -8.992686e-04
## [66,] 5.342271e-03
##
## $log_evidence
## [1] -160.9409
##
## $converge
## [1] "YES"
##
## $iter_counts
## [1] 148
Use the viz_post_coefs() function to visualize
the posterior coefficient summaries for model 3 and model 6, based on
the strong prior specification.
### add more code chunks if you like
viz_post_coefs(laplace_03_strong$mode[-length(laplace_03_strong$mode)],
sqrt(diag(laplace_03_strong$var_matrix))[-length(laplace_03_strong$mode)],
info_03_strong$design_matrix %>% colnames())
viz_post_coefs(laplace_06_strong$mode[-length(laplace_06_strong$mode)],
sqrt(diag(laplace_06_strong$var_matrix))[-length(laplace_06_strong$mode)],
info_06_strong$design_matrix %>% colnames())
You will fit one more set of Bayesian models with a very strong prior on the regression coefficients. The prior standard deviation will be equal to 1/50.
Complete the first code chunk below, which defines the list of required information for both the model 3 and model 6 formulations using the very strong prior on the regression coefficients. All other information, data and the \(\sigma\) prior, are the same as before.
Run the Laplace Approximation using the strong prior for both
the model 3 and model 6 formulations. Assign the results to
laplace_03_very_strong and
laplace_06_very_strong.
Confirm that the optimizations converged for both laplace approximation results.
info_03_very_strong <- list(
yobs = df$y,
design_matrix = info_03_weak$design_matrix,
mu_beta = 0,
tau_beta = 1/50,
sigma_rate = 1
)
info_06_very_strong <- list(
yobs = df$y,
design_matrix = info_06_weak$design_matrix,
mu_beta = 0,
tau_beta = 1/50,
sigma_rate = 1
)
Execute the Laplace Approximation.
### add more code chunks if you like
laplace_03_very_strong <- my_laplace(start_guess_03,
lm_logpost,
info_03_very_strong)
laplace_03_very_strong
## $mode
## [1] 0.003592761 0.006873296 -0.003563924 -0.005548833 -0.034591615
## [6] -0.005596963 0.006696281 -0.010922248 -0.036247920 -0.082665953
##
## $var_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 3.845484e-04 2.590456e-07 -1.379063e-05 -2.342351e-07 -1.718064e-05
## [2,] 2.590456e-07 3.840058e-04 3.516814e-06 -1.839586e-06 -3.016308e-06
## [3,] -1.379063e-05 3.516814e-06 3.644930e-04 -1.862478e-06 -1.159401e-05
## [4,] -2.342351e-07 -1.839586e-06 -1.862478e-06 3.837901e-04 2.426882e-06
## [5,] -1.718064e-05 -3.016308e-06 -1.159401e-05 2.426882e-06 3.747607e-04
## [6,] -9.397074e-07 -3.102062e-06 -3.791051e-06 2.222885e-06 -5.962896e-06
## [7,] 1.343448e-06 -1.574857e-05 6.109602e-06 -9.945583e-06 2.864426e-06
## [8,] -3.066295e-06 -6.377984e-06 -2.351645e-06 -1.551120e-05 6.543729e-07
## [9,] -1.471935e-05 2.996232e-06 -3.702900e-05 -2.145655e-06 -2.186027e-05
## [10,] -6.498839e-05 -6.513481e-05 -1.287717e-05 5.111098e-05 3.033539e-04
## [,6] [,7] [,8] [,9] [,10]
## [1,] -9.397074e-07 1.343448e-06 -3.066295e-06 -1.471935e-05 -6.498839e-05
## [2,] -3.102062e-06 -1.574857e-05 -6.377984e-06 2.996232e-06 -6.513481e-05
## [3,] -3.791051e-06 6.109602e-06 -2.351645e-06 -3.702900e-05 -1.287717e-05
## [4,] 2.222885e-06 -9.945583e-06 -1.551120e-05 -2.145655e-06 5.111098e-05
## [5,] -5.962896e-06 2.864426e-06 6.543729e-07 -2.186027e-05 3.033539e-04
## [6,] 3.826461e-04 -6.989653e-06 9.383101e-06 -1.666292e-05 3.746314e-05
## [7,] -6.989653e-06 3.529481e-04 -2.161346e-05 8.768168e-06 -5.358871e-05
## [8,] 9.383101e-06 -2.161346e-05 3.545232e-04 2.842741e-06 1.034923e-04
## [9,] -1.666292e-05 8.768168e-06 2.842741e-06 3.207896e-04 2.649717e-04
## [10,] 3.746314e-05 -5.358871e-05 1.034923e-04 2.649717e-04 5.546760e-03
##
## $log_evidence
## [1] -140.2491
##
## $converge
## [1] "YES"
##
## $iter_counts
## [1] 84
laplace_06_very_strong <- my_laplace(start_guess_06,
lm_logpost,
info_06_very_strong)
laplace_06_very_strong
## $mode
## [1] 9.595320e-03 2.025166e-04 4.505341e-04 1.246627e-03 2.045638e-03
## [6] 1.145230e-03 2.340198e-04 1.992312e-03 1.798180e-03 9.429094e-04
## [11] 1.639782e-03 -8.279119e-05 2.350166e-04 -7.058068e-03 -1.630164e-02
## [16] -1.277296e-02 -5.098678e-02 -8.853261e-04 9.585182e-04 -1.877547e-04
## [21] -4.061852e-04 2.894994e-04 -2.911555e-04 -1.957401e-03 -1.220925e-03
## [26] -1.859333e-04 5.269584e-05 -1.752511e-03 3.444512e-04 -9.019286e-05
## [31] -4.984301e-04 -1.634918e-04 -3.140493e-03 -2.410296e-03 -1.818257e-03
## [36] -7.205437e-04 3.737394e-05 -5.742500e-04 -2.464914e-04 -2.426018e-03
## [41] 3.902126e-04 -3.532866e-03 3.818524e-03 -3.519940e-04 8.338765e-04
## [46] 1.454893e-03 -1.922879e-03 -2.627591e-03 -1.208161e-03 -1.183203e-03
## [51] -7.897388e-04 -4.436582e-03 7.269596e-04 5.313389e-03 6.979266e-04
## [56] -9.424258e-04 -1.477504e-02 -1.059741e-02 -5.655724e-03 -3.734206e-03
## [61] -1.390676e-03 -2.459181e-03 -2.553076e-03 -6.011259e-03 1.950180e-04
## [66] -1.901450e-01
##
## $var_matrix
## [,1] [,2] [,3] [,4] [,5]
## [1,] 3.837176e-04 -1.518928e-06 -1.564506e-06 -1.076522e-06 -1.257067e-06
## [2,] -1.518928e-06 3.991733e-04 -4.444459e-07 -1.235300e-08 5.998136e-08
## [3,] -1.564506e-06 -4.444459e-07 3.989233e-04 -3.275556e-07 2.570194e-08
## [4,] -1.076522e-06 -1.235300e-08 -3.275556e-07 3.993239e-04 -3.817015e-07
## [5,] -1.257067e-06 5.998136e-08 2.570194e-08 -3.817015e-07 3.989651e-04
## [6,] -1.109564e-06 4.151158e-08 4.573016e-08 4.067442e-08 -3.591925e-07
## [7,] -1.249922e-06 2.338736e-08 2.507771e-08 2.673983e-08 3.746780e-09
## [8,] -1.833931e-06 3.847787e-08 4.992998e-08 7.859778e-08 1.358024e-07
## [9,] -1.249185e-06 2.653786e-08 3.665102e-08 6.375024e-08 1.106793e-07
## [10,] -1.207047e-06 3.376781e-08 4.032391e-08 5.274356e-08 8.853751e-08
## [11,] -2.984415e-07 1.809009e-07 4.782518e-08 6.883375e-08 1.173174e-07
## [12,] -1.676348e-06 -3.308032e-07 1.673559e-08 2.449224e-08 3.773776e-08
## [13,] 3.837901e-07 2.072744e-07 5.203185e-09 2.883411e-09 5.111411e-09
## [14,] -8.551570e-07 -6.831185e-08 2.558705e-07 -3.131077e-07 -8.770093e-08
## [15,] -1.289480e-05 -7.653976e-07 -9.777198e-07 -9.141577e-07 -1.632436e-06
## [16,] -5.381842e-06 3.676791e-07 -2.667817e-07 -4.498030e-07 3.738204e-07
## [17,] -1.674485e-05 -1.778064e-06 -1.544158e-06 -1.242333e-06 -2.595546e-06
## [18,] -5.749400e-07 -6.385854e-08 2.976913e-08 -4.156176e-08 -1.102950e-07
## [19,] 6.688739e-07 1.100974e-07 2.470222e-07 3.683119e-08 3.928000e-08
## [20,] -1.666892e-07 8.784905e-09 4.948827e-10 -1.319863e-07 -7.649760e-08
## [21,] -1.575422e-07 -4.656215e-08 -4.758724e-08 -1.064912e-07 6.655866e-08
## [22,] 2.134833e-07 -2.217272e-08 -1.356755e-08 -7.477695e-09 1.454313e-07
## [23,] -1.467417e-07 6.150377e-09 7.643870e-09 5.032730e-09 1.329346e-08
## [24,] 1.795823e-07 -8.321694e-09 -1.706529e-08 -4.728880e-08 -8.927159e-08
## [25,] 2.112884e-07 -6.419615e-09 -1.238029e-08 -3.167199e-08 -5.811371e-08
## [26,] 2.304404e-07 3.658512e-08 3.793090e-08 3.331888e-08 5.259152e-08
## [27,] 1.415656e-07 7.250357e-08 4.695195e-08 4.325408e-08 7.282597e-08
## [28,] -8.426327e-07 -6.208568e-08 -1.043983e-09 -2.613577e-08 -5.767323e-08
## [29,] 6.627573e-08 4.841503e-08 1.017957e-08 1.242478e-08 2.508919e-08
## [30,] -2.061934e-08 -6.363355e-07 -1.688978e-07 7.497963e-08 2.556779e-07
## [31,] -6.933501e-07 -2.509352e-07 -9.713017e-07 -2.224666e-07 1.283729e-07
## [32,] -6.555488e-07 1.370992e-08 -2.455097e-07 -4.315758e-07 -3.437790e-07
## [33,] -1.679837e-06 1.197064e-07 7.258005e-08 -4.056090e-07 -1.264772e-06
## [34,] -1.648317e-06 7.301378e-08 5.208042e-08 -1.050214e-08 -4.871504e-07
## [35,] -1.488793e-06 2.991577e-08 2.125771e-08 -1.252219e-08 -5.861372e-08
## [36,] -1.485382e-06 2.932432e-08 2.714374e-08 1.059529e-08 1.462491e-08
## [37,] -6.110354e-07 1.763931e-08 1.867346e-08 1.694071e-08 2.823296e-08
## [38,] -4.960866e-07 7.916099e-08 8.089889e-08 6.496812e-08 1.036155e-07
## [39,] -6.117359e-08 1.846734e-07 7.660015e-08 6.633759e-08 1.045342e-07
## [40,] -1.422390e-06 -2.114870e-07 4.939768e-08 5.758143e-09 4.609202e-09
## [41,] 2.121388e-07 1.470179e-07 -9.387142e-09 -3.158845e-09 -8.085157e-09
## [42,] -3.385292e-06 2.224860e-07 -3.589126e-07 -3.513635e-07 -7.246068e-07
## [43,] 1.595862e-06 2.573059e-07 6.559765e-08 8.918269e-08 1.811941e-07
## [44,] -6.455058e-08 -3.008161e-09 -5.889163e-08 -1.107162e-07 -6.295850e-09
## [45,] -3.812199e-08 -1.882367e-07 -1.553674e-07 -9.357741e-08 4.525079e-07
## [46,] 5.648680e-07 -8.474648e-08 -5.261757e-08 -3.402557e-08 5.002210e-07
## [47,] -5.661182e-07 1.430628e-08 1.169984e-08 -1.762835e-08 -1.592823e-08
## [48,] -4.664659e-07 1.326195e-08 3.605053e-09 -3.950586e-08 -8.093377e-08
## [49,] -9.888857e-08 1.076593e-08 6.293541e-09 -1.424979e-08 -3.045218e-08
## [50,] -7.405878e-08 1.558495e-07 1.591564e-07 1.286055e-07 2.046428e-07
## [51,] 2.067333e-07 2.227522e-07 1.948957e-07 1.625305e-07 2.737586e-07
## [52,] -2.037791e-06 -7.381924e-08 -9.530580e-09 -7.110718e-08 -1.640797e-07
## [53,] 3.406248e-07 8.501150e-08 4.121024e-08 4.217995e-08 8.734880e-08
## [54,] 6.652979e-06 -2.006045e-06 3.072726e-07 8.613631e-07 2.060142e-06
## [55,] 5.872295e-07 -6.770527e-07 -2.748992e-06 -3.448906e-07 8.681840e-07
## [56,] -1.225304e-06 7.373955e-08 -6.446816e-07 -9.707759e-07 -8.946744e-07
## [57,] -5.017787e-06 4.354222e-07 2.709191e-07 -1.173723e-06 -3.797297e-06
## [58,] -4.220555e-06 2.367368e-07 1.470570e-07 -8.086236e-08 -1.500183e-06
## [59,] -2.988305e-06 7.223578e-08 4.567180e-08 -5.569282e-08 -1.748094e-07
## [60,] -2.408010e-06 5.281503e-08 3.811824e-08 -3.150115e-08 -6.805816e-08
## [61,] -9.085531e-07 3.640897e-08 3.188060e-08 4.302856e-09 1.141384e-09
## [62,] -4.706613e-07 3.231449e-07 3.295320e-07 2.645382e-07 4.228842e-07
## [63,] -3.944244e-07 5.704071e-07 2.721275e-07 2.190975e-07 3.272147e-07
## [64,] -1.870679e-06 -4.237973e-07 2.945506e-07 1.477507e-07 2.671009e-07
## [65,] -2.623744e-07 3.281596e-07 -1.038818e-07 -8.064689e-08 -1.633992e-07
## [66,] -1.260489e-04 -3.626851e-06 -6.310892e-06 -1.492951e-05 -2.623075e-05
## [,6] [,7] [,8] [,9] [,10]
## [1,] -1.109564e-06 -1.249922e-06 -1.833931e-06 -1.249185e-06 -1.207047e-06
## [2,] 4.151158e-08 2.338736e-08 3.847787e-08 2.653786e-08 3.376781e-08
## [3,] 4.573016e-08 2.507771e-08 4.992998e-08 3.665102e-08 4.032391e-08
## [4,] 4.067442e-08 2.673983e-08 7.859778e-08 6.375024e-08 5.274356e-08
## [5,] -3.591925e-07 3.746780e-09 1.358024e-07 1.106793e-07 8.853751e-08
## [6,] 3.992703e-04 -3.651945e-07 2.220361e-08 6.797898e-08 5.640197e-08
## [7,] -3.651945e-07 3.993192e-04 -3.689930e-07 -1.912230e-09 2.951570e-08
## [8,] 2.220361e-08 -3.689930e-07 3.986087e-04 -5.014213e-07 3.591678e-08
## [9,] 6.797898e-08 -1.912230e-09 -5.014213e-07 3.992335e-04 -2.950465e-07
## [10,] 5.640197e-08 2.951570e-08 3.591678e-08 -2.950465e-07 3.992294e-04
## [11,] 7.150613e-08 3.074398e-08 9.557878e-08 5.271740e-08 -3.728258e-07
## [12,] 2.739159e-08 2.264042e-08 2.861010e-08 9.967844e-09 -1.258520e-07
## [13,] 1.895142e-09 -1.451909e-09 5.527929e-09 1.086818e-08 4.179298e-08
## [14,] 9.149868e-08 -2.122490e-07 3.379792e-07 2.604244e-07 -1.477561e-07
## [15,] -1.413283e-06 -1.072105e-06 -1.546715e-06 -8.664735e-07 -8.539284e-07
## [16,] 2.948725e-07 -5.040501e-07 -6.187050e-07 -3.996449e-07 -1.520050e-06
## [17,] -1.698550e-06 -8.594748e-07 -1.176841e-06 -7.766390e-07 -1.597800e-06
## [18,] -7.182063e-08 -1.870809e-08 -5.463117e-08 -4.327813e-08 -3.028556e-08
## [19,] 2.479168e-08 1.740528e-08 5.038281e-08 4.200075e-08 5.070093e-08
## [20,] -8.412429e-09 4.805712e-09 -3.181646e-09 -4.450111e-09 1.088428e-08
## [21,] 1.105869e-07 -9.767305e-09 -4.769298e-08 -3.685754e-08 -1.900740e-08
## [22,] 1.733709e-07 -4.357470e-08 -2.321355e-08 -2.399703e-10 6.592876e-09
## [23,] -3.272623e-08 -1.736834e-07 -4.130813e-08 5.619199e-09 1.586510e-08
## [24,] -7.152441e-08 -5.149901e-08 3.155614e-07 1.410936e-07 -2.242083e-08
## [25,] -3.399790e-08 7.772968e-11 1.598568e-07 1.585318e-07 2.066411e-08
## [26,] 3.556279e-08 2.884986e-08 4.141242e-08 6.360896e-08 -5.300243e-08
## [27,] 4.843826e-08 3.254177e-08 4.746746e-08 3.420047e-08 -6.824914e-08
## [28,] -3.075804e-08 7.837583e-09 -4.677204e-08 -4.646579e-08 -4.843254e-08
## [29,] 1.557847e-08 3.936364e-09 1.674782e-08 1.418938e-08 1.731460e-08
## [30,] 1.755251e-07 8.258053e-08 1.043056e-07 6.302207e-08 1.144535e-07
## [31,] 9.982544e-08 6.480777e-08 7.048046e-08 3.882008e-08 9.724019e-08
## [32,] 4.574408e-08 3.700744e-08 4.909661e-08 3.103637e-08 5.464245e-08
## [33,] -4.400080e-07 2.055843e-08 -8.151899e-09 -3.028220e-08 3.104888e-08
## [34,] -8.385078e-07 -4.893843e-07 -8.283728e-08 -3.214399e-08 1.347222e-08
## [35,] -5.306391e-07 -7.171173e-07 -2.813233e-07 -5.765006e-08 6.762329e-09
## [36,] -4.935546e-08 -2.292959e-07 -1.044457e-06 -3.581262e-07 1.729536e-08
## [37,] 2.034854e-08 -6.139984e-09 -3.355641e-07 -2.736634e-07 -1.291621e-07
## [38,] 7.236140e-08 6.052649e-08 5.925803e-08 -1.117344e-07 -6.461731e-07
## [39,] 7.054240e-08 5.701671e-08 7.268306e-08 2.981553e-08 -3.099615e-07
## [40,] 1.611696e-08 3.950314e-08 -2.460865e-08 -4.478314e-08 -1.050269e-07
## [41,] -7.937082e-09 -6.723492e-09 3.536067e-09 1.028254e-08 4.031094e-08
## [42,] -4.828625e-07 -1.604329e-07 -3.245144e-07 -2.370381e-07 -2.383833e-07
## [43,] 9.461212e-08 7.349475e-08 2.115852e-07 1.750903e-07 2.093632e-07
## [44,] -3.095669e-08 1.471234e-08 -3.661455e-09 -8.066384e-09 3.964872e-08
## [45,] 3.854305e-07 -6.169216e-08 -1.074642e-07 -6.921501e-08 -5.029490e-08
## [46,] 4.609527e-07 -1.599959e-07 -4.379038e-08 7.614437e-09 1.453747e-08
## [47,] -1.399748e-07 -3.626144e-07 -1.512954e-07 -5.047201e-08 1.703735e-08
## [48,] -8.000707e-08 -9.988733e-08 -2.459030e-07 -5.082716e-08 -1.872062e-09
## [49,] -1.440266e-08 2.729362e-09 -6.946297e-09 1.894328e-08 -1.179482e-07
## [50,] 1.418476e-07 1.190451e-07 1.340431e-07 -4.289829e-08 -8.234870e-07
## [51,] 1.871179e-07 1.351514e-07 1.639790e-07 8.418713e-08 -3.737835e-07
## [52,] -9.120175e-08 2.134079e-08 -1.126813e-07 -1.189492e-07 -2.045003e-07
## [53,] 5.656584e-08 1.691555e-08 4.593442e-08 4.079599e-08 9.049473e-08
## [54,] 1.382170e-06 5.706166e-07 9.097605e-07 6.159209e-07 8.507702e-07
## [55,] 5.904268e-07 3.370151e-07 4.634472e-07 2.965651e-07 5.510101e-07
## [56,] 1.714829e-07 1.269990e-07 1.554324e-07 9.427360e-08 1.872756e-07
## [57,] -1.273181e-06 1.189169e-07 -1.687557e-07 -2.337975e-07 2.203266e-08
## [58,] -1.992285e-06 -1.002151e-06 -2.956831e-07 -2.035137e-07 -1.916285e-08
## [59,] -1.149911e-06 -1.396583e-06 -4.666268e-07 -1.623605e-07 3.285308e-09
## [60,] -1.513434e-07 -3.139658e-07 -1.624495e-06 -5.618642e-07 1.720695e-08
## [61,] 1.001888e-08 -4.279685e-09 -4.881638e-07 -3.024492e-07 -2.900360e-07
## [62,] 2.937690e-07 2.454011e-07 2.658591e-07 -1.575453e-07 -1.871266e-06
## [63,] 2.270473e-07 2.190376e-07 2.298862e-07 9.376540e-08 -9.292369e-07
## [64,] 2.132269e-07 1.846034e-07 5.532938e-08 -3.538897e-08 -2.251384e-07
## [65,] -1.137058e-07 -5.224511e-08 -6.269253e-08 -2.645135e-08 7.301313e-08
## [66,] -1.526080e-05 -4.299249e-06 -2.218816e-05 -1.939079e-05 -1.241507e-05
## [,11] [,12] [,13] [,14] [,15]
## [1,] -2.984415e-07 -1.676348e-06 3.837901e-07 -8.551570e-07 -1.289480e-05
## [2,] 1.809009e-07 -3.308032e-07 2.072744e-07 -6.831185e-08 -7.653976e-07
## [3,] 4.782518e-08 1.673559e-08 5.203185e-09 2.558705e-07 -9.777198e-07
## [4,] 6.883375e-08 2.449224e-08 2.883411e-09 -3.131077e-07 -9.141577e-07
## [5,] 1.173174e-07 3.773776e-08 5.111411e-09 -8.770093e-08 -1.632436e-06
## [6,] 7.150613e-08 2.739159e-08 1.895142e-09 9.149868e-08 -1.413283e-06
## [7,] 3.074398e-08 2.264042e-08 -1.451909e-09 -2.122490e-07 -1.072105e-06
## [8,] 9.557878e-08 2.861010e-08 5.527929e-09 3.379792e-07 -1.546715e-06
## [9,] 5.271740e-08 9.967844e-09 1.086818e-08 2.604244e-07 -8.664735e-07
## [10,] -3.728258e-07 -1.258520e-07 4.179298e-08 -1.477561e-07 -8.539284e-07
## [11,] 3.993314e-04 -7.433355e-08 -1.332662e-07 -4.034124e-07 -6.337085e-07
## [12,] -7.433355e-08 3.993620e-04 1.350380e-07 -5.446653e-07 -1.024964e-06
## [13,] -1.332662e-07 1.350380e-07 3.996269e-04 -6.692764e-08 1.364628e-07
## [14,] -4.034124e-07 -5.446653e-07 -6.692764e-08 3.854309e-04 -5.794746e-07
## [15,] -6.337085e-07 -1.024964e-06 1.364628e-07 -5.794746e-07 3.802011e-04
## [16,] -1.685961e-06 -1.190673e-06 -6.154080e-08 -3.378624e-05 -6.454426e-06
## [17,] -1.716523e-06 -1.243706e-06 1.264408e-07 -8.795460e-07 -5.313456e-05
## [18,] -7.182627e-09 -8.911771e-08 4.789511e-08 -1.161073e-06 -3.583002e-07
## [19,] 7.057768e-08 2.430726e-08 6.508775e-09 -1.492129e-06 2.139273e-07
## [20,] 9.363298e-09 1.452141e-08 -7.451097e-10 -9.847599e-07 5.634852e-10
## [21,] -2.482432e-08 -3.008607e-09 1.541160e-09 -1.760344e-06 6.145287e-07
## [22,] 1.242016e-08 3.617393e-09 3.569725e-09 -1.607041e-06 4.635975e-07
## [23,] 1.403183e-08 1.926206e-08 -1.462198e-09 -1.256265e-06 -2.052075e-07
## [24,] -5.483754e-08 9.562436e-09 -8.457989e-09 -1.253411e-06 3.236518e-07
## [25,] -3.563949e-08 3.819007e-09 -5.152012e-09 -5.728108e-07 2.775638e-07
## [26,] -7.382049e-08 -1.417487e-09 6.165407e-09 -9.836703e-07 -6.433304e-07
## [27,] -2.594696e-07 -9.930537e-09 -5.158155e-08 -6.017211e-07 -6.802591e-07
## [28,] -8.236903e-08 -2.451688e-07 -8.277902e-09 -1.196068e-06 -7.068608e-07
## [29,] -3.671565e-08 1.560114e-09 -2.166817e-07 2.076720e-07 -5.368592e-08
## [30,] 2.172423e-07 -1.628440e-07 1.372832e-07 1.847612e-07 -8.153230e-07
## [31,] 9.168151e-08 8.581286e-08 -8.493245e-09 3.318583e-07 -2.023100e-06
## [32,] 5.259082e-08 5.002684e-08 -6.532784e-09 -1.234095e-07 -1.640200e-06
## [33,] -1.078186e-08 7.232058e-08 -2.455242e-08 1.134532e-06 -2.840231e-06
## [34,] -2.093903e-08 4.962202e-08 -1.812173e-08 9.066528e-07 -2.253625e-06
## [35,] -2.034931e-08 3.900340e-08 -1.211695e-08 -2.169467e-07 -1.525419e-06
## [36,] 1.180049e-08 3.562465e-08 -6.956096e-09 -2.395415e-08 -1.467374e-06
## [37,] 4.704707e-09 1.416302e-08 2.324672e-09 2.813712e-08 -6.800692e-07
## [38,] -3.130160e-07 -4.771858e-08 3.824583e-08 -1.063662e-06 -1.577413e-06
## [39,] -6.021475e-07 -3.207234e-08 -1.325381e-07 -1.152495e-06 -1.307782e-06
## [40,] -1.179738e-07 -5.097186e-07 8.261071e-08 -9.642735e-07 -1.631076e-06
## [41,] -1.281470e-07 8.836376e-08 -3.316838e-07 -9.967221e-08 2.000906e-07
## [42,] -2.170094e-07 -2.793204e-07 1.046124e-07 -3.254780e-06 -1.536366e-06
## [43,] 2.770262e-07 1.252991e-07 1.281636e-08 -4.418504e-06 -5.735761e-07
## [44,] 3.814294e-08 4.644841e-08 -1.531032e-09 -2.442520e-06 1.358640e-07
## [45,] -4.112565e-08 -3.526280e-08 1.801610e-08 -4.938748e-06 2.612220e-06
## [46,] 3.725663e-08 -2.149155e-09 1.422857e-08 -3.717637e-06 1.498570e-06
## [47,] -5.636272e-09 4.927078e-08 -9.861763e-09 -2.141211e-06 -5.133265e-07
## [48,] -4.042478e-08 4.159548e-08 -1.364624e-08 -1.685513e-06 -5.007667e-07
## [49,] -3.165773e-08 1.875572e-08 -1.443093e-09 -7.352686e-07 -2.441193e-07
## [50,] -4.032132e-07 -2.051708e-08 5.187424e-08 -2.509340e-06 -2.716963e-06
## [51,] -8.617700e-07 -1.276670e-07 -1.784669e-07 -2.034879e-06 -2.655874e-06
## [52,] -3.840319e-07 -5.533531e-07 1.191914e-08 -2.644848e-06 -2.159946e-06
## [53,] -1.328485e-07 4.021466e-08 -3.522331e-07 4.757786e-07 2.323134e-08
## [54,] 1.154371e-06 1.662452e-08 2.677945e-07 3.137190e-06 -4.061235e-07
## [55,] 5.753613e-07 4.309812e-07 -3.116697e-08 -2.836289e-07 -6.546366e-06
## [56,] 1.772485e-07 1.768894e-07 -2.490897e-08 -1.779113e-07 -4.984798e-06
## [57,] -1.756896e-07 2.381085e-07 -1.050847e-07 5.694711e-06 -8.865390e-06
## [58,] -1.657065e-07 1.500923e-07 -7.112202e-08 3.509633e-06 -5.811213e-06
## [59,] -7.666193e-08 1.031000e-07 -3.467764e-08 -7.032931e-07 -3.223422e-06
## [60,] -3.615107e-08 8.411383e-08 -2.399241e-08 -9.517121e-07 -2.566731e-06
## [61,] -3.439385e-08 3.891545e-08 1.232394e-09 -5.380796e-07 -1.278984e-06
## [62,] -9.054774e-07 -6.092530e-08 1.262812e-07 -4.556676e-06 -5.646235e-06
## [63,] -1.834886e-06 -2.615104e-07 -3.498364e-07 -4.847940e-06 -5.345072e-06
## [64,] -4.834754e-07 -1.001870e-06 1.484420e-07 -2.461344e-06 -4.269794e-06
## [65,] -4.107026e-07 1.344665e-07 -5.748620e-07 -3.900477e-07 3.013515e-07
## [66,] -1.998256e-05 -1.396871e-06 -2.320301e-06 5.702419e-05 8.021419e-05
## [,16] [,17] [,18] [,19] [,20]
## [1,] -5.381842e-06 -1.674485e-05 -5.749400e-07 6.688739e-07 -1.666892e-07
## [2,] 3.676791e-07 -1.778064e-06 -6.385854e-08 1.100974e-07 8.784905e-09
## [3,] -2.667817e-07 -1.544158e-06 2.976913e-08 2.470222e-07 4.948827e-10
## [4,] -4.498030e-07 -1.242333e-06 -4.156176e-08 3.683119e-08 -1.319863e-07
## [5,] 3.738204e-07 -2.595546e-06 -1.102950e-07 3.928000e-08 -7.649760e-08
## [6,] 2.948725e-07 -1.698550e-06 -7.182063e-08 2.479168e-08 -8.412429e-09
## [7,] -5.040501e-07 -8.594748e-07 -1.870809e-08 1.740528e-08 4.805712e-09
## [8,] -6.187050e-07 -1.176841e-06 -5.463117e-08 5.038281e-08 -3.181646e-09
## [9,] -3.996449e-07 -7.766390e-07 -4.327813e-08 4.200075e-08 -4.450111e-09
## [10,] -1.520050e-06 -1.597800e-06 -3.028556e-08 5.070093e-08 1.088428e-08
## [11,] -1.685961e-06 -1.716523e-06 -7.182627e-09 7.057768e-08 9.363298e-09
## [12,] -1.190673e-06 -1.243706e-06 -8.911771e-08 2.430726e-08 1.452141e-08
## [13,] -6.154080e-08 1.264408e-07 4.789511e-08 6.508775e-09 -7.451097e-10
## [14,] -3.378624e-05 -8.795460e-07 -1.161073e-06 -1.492129e-06 -9.847599e-07
## [15,] -6.454426e-06 -5.313456e-05 -3.583002e-07 2.139273e-07 5.634852e-10
## [16,] 2.689079e-04 1.190234e-05 -4.272723e-06 -4.494377e-06 -2.338627e-06
## [17,] 1.190234e-05 1.406736e-04 3.308131e-06 -7.499128e-07 -9.307782e-08
## [18,] -4.272723e-06 3.308131e-06 3.992502e-04 -3.969804e-07 -1.517053e-08
## [19,] -4.494377e-06 -7.499128e-07 -3.969804e-07 3.989168e-04 -2.852384e-07
## [20,] -2.338627e-06 -9.307782e-08 -1.517053e-08 -2.852384e-07 3.995279e-04
## [21,] -4.472386e-06 2.286938e-06 1.573243e-07 4.318534e-08 -4.161710e-07
## [22,] -3.196252e-06 9.363151e-07 8.617832e-08 6.330585e-08 1.066809e-08
## [23,] -1.959613e-06 -5.973641e-07 2.813052e-08 3.770594e-08 2.386630e-08
## [24,] -1.288802e-06 -1.773541e-07 5.387485e-08 -7.491110e-09 2.453526e-08
## [25,] -5.288036e-07 -5.095393e-08 2.965669e-08 -1.084326e-08 1.203628e-08
## [26,] -2.971297e-06 -2.719219e-06 -1.354727e-08 6.520777e-08 3.104110e-08
## [27,] -2.551129e-06 -3.079801e-06 9.326881e-08 7.156066e-08 2.494567e-08
## [28,] -2.918888e-06 -7.076084e-07 -2.490361e-07 2.347962e-09 3.883211e-08
## [29,] 5.769754e-07 -4.019227e-07 1.563958e-07 1.083509e-08 -7.431937e-09
## [30,] 5.807124e-06 -1.158867e-05 3.766824e-07 1.834484e-07 -4.702023e-08
## [31,] -1.547125e-06 -6.916259e-06 -1.885821e-08 3.012817e-08 -5.025285e-08
## [32,] -7.231620e-07 -3.180963e-06 -1.040821e-07 -5.148456e-08 -1.052054e-07
## [33,] 4.482046e-06 -6.555536e-06 -2.196557e-07 -1.092092e-07 5.697047e-09
## [34,] 2.445425e-06 -3.870423e-06 -1.209271e-07 -7.155106e-08 -1.905936e-08
## [35,] -1.173710e-06 -1.618253e-06 2.812004e-09 -6.499181e-09 1.996509e-08
## [36,] -1.347727e-06 -1.389005e-06 -4.543815e-09 1.424805e-08 1.656867e-08
## [37,] -8.007801e-07 -8.778874e-07 -1.090199e-08 1.752179e-08 7.466427e-09
## [38,] -5.711723e-06 -5.475671e-06 -3.321743e-08 1.166221e-07 5.845881e-08
## [39,] -6.247480e-06 -4.990344e-06 1.621324e-08 1.208133e-07 6.103419e-08
## [40,] -2.684738e-06 -4.528877e-06 -1.102434e-07 4.853672e-08 4.134035e-08
## [41,] -4.859037e-07 1.067277e-06 7.642330e-08 -2.909645e-09 1.784325e-09
## [42,] -2.537992e-05 2.662227e-05 -2.794083e-06 -1.363641e-06 4.344068e-08
## [43,] -1.991351e-05 -1.894762e-06 -1.453744e-06 -3.391793e-06 -7.541425e-07
## [44,] -8.153056e-06 -3.413739e-07 -6.292478e-08 -7.975880e-07 -1.110424e-06
## [45,] -1.634376e-05 9.751620e-06 5.714363e-07 2.223115e-07 -1.138912e-06
## [46,] -1.021831e-05 4.035352e-06 2.899926e-07 1.982306e-07 3.952873e-08
## [47,] -4.771502e-06 -1.613704e-06 7.871412e-08 6.406386e-08 6.050796e-08
## [48,] -3.286957e-06 -1.561851e-06 6.293847e-08 2.208698e-08 4.762152e-08
## [49,] -1.770091e-06 -1.095286e-06 2.352448e-08 1.593752e-08 2.404978e-08
## [50,] -1.182860e-05 -1.125308e-05 -6.451773e-08 2.440870e-07 1.214071e-07
## [51,] -1.030370e-05 -1.357532e-05 2.142286e-07 2.801647e-07 1.050937e-07
## [52,] -1.119342e-05 -9.010960e-07 -6.592187e-07 1.157292e-08 1.289198e-07
## [53,] 2.761807e-06 -2.495442e-06 4.052194e-07 4.095682e-08 -2.841969e-08
## [54,] 5.044270e-05 -8.426531e-05 4.825500e-06 1.025353e-06 -5.464631e-07
## [55,] -6.277374e-06 -3.788468e-05 1.823939e-07 -1.513762e-06 -6.218474e-07
## [56,] -2.673772e-06 -1.194017e-05 -4.986030e-07 -6.540151e-07 -5.996019e-08
## [57,] 2.142614e-05 -2.470218e-05 -8.681542e-07 -5.299646e-07 4.309626e-07
## [58,] 1.051231e-05 -1.359724e-05 -4.245507e-07 -3.244098e-07 -5.906517e-08
## [59,] -3.899810e-06 -4.630767e-06 2.690554e-08 -1.629766e-08 6.385114e-08
## [60,] -4.279987e-06 -3.365368e-06 4.264741e-08 2.007096e-08 6.041758e-08
## [61,] -2.815932e-06 -2.423009e-06 7.057593e-09 3.425739e-08 3.433293e-08
## [62,] -2.404563e-05 -2.317509e-05 -1.407621e-07 4.931705e-07 2.453924e-07
## [63,] -2.819765e-05 -1.902918e-05 -2.593843e-07 4.419366e-07 2.794607e-07
## [64,] -4.956157e-06 -2.326748e-05 3.030272e-07 3.387291e-07 9.304241e-08
## [65,] -4.255208e-06 7.821312e-06 -1.506008e-07 -1.051856e-07 3.257159e-08
## [66,] 9.057870e-05 1.570613e-04 1.018219e-05 -1.238544e-05 1.297754e-06
## [,21] [,22] [,23] [,24] [,25]
## [1,] -1.575422e-07 2.134833e-07 -1.467417e-07 1.795823e-07 2.112884e-07
## [2,] -4.656215e-08 -2.217272e-08 6.150377e-09 -8.321694e-09 -6.419615e-09
## [3,] -4.758724e-08 -1.356755e-08 7.643870e-09 -1.706529e-08 -1.238029e-08
## [4,] -1.064912e-07 -7.477695e-09 5.032730e-09 -4.728880e-08 -3.167199e-08
## [5,] 6.655866e-08 1.454313e-07 1.329346e-08 -8.927159e-08 -5.811371e-08
## [6,] 1.105869e-07 1.733709e-07 -3.272623e-08 -7.152441e-08 -3.399790e-08
## [7,] -9.767305e-09 -4.357470e-08 -1.736834e-07 -5.149901e-08 7.772968e-11
## [8,] -4.769298e-08 -2.321355e-08 -4.130813e-08 3.155614e-07 1.598568e-07
## [9,] -3.685754e-08 -2.399703e-10 5.619199e-09 1.410936e-07 1.585318e-07
## [10,] -1.900740e-08 6.592876e-09 1.586510e-08 -2.242083e-08 2.066411e-08
## [11,] -2.482432e-08 1.242016e-08 1.403183e-08 -5.483754e-08 -3.563949e-08
## [12,] -3.008607e-09 3.617393e-09 1.926206e-08 9.562436e-09 3.819007e-09
## [13,] 1.541160e-09 3.569725e-09 -1.462198e-09 -8.457989e-09 -5.152012e-09
## [14,] -1.760344e-06 -1.607041e-06 -1.256265e-06 -1.253411e-06 -5.728108e-07
## [15,] 6.145287e-07 4.635975e-07 -2.052075e-07 3.236518e-07 2.775638e-07
## [16,] -4.472386e-06 -3.196252e-06 -1.959613e-06 -1.288802e-06 -5.288036e-07
## [17,] 2.286938e-06 9.363151e-07 -5.973641e-07 -1.773541e-07 -5.095393e-08
## [18,] 1.573243e-07 8.617832e-08 2.813052e-08 5.387485e-08 2.965669e-08
## [19,] 4.318534e-08 6.330585e-08 3.770594e-08 -7.491110e-09 -1.084326e-08
## [20,] -4.161710e-07 1.066809e-08 2.386630e-08 2.453526e-08 1.203628e-08
## [21,] 3.986989e-04 -4.658724e-07 -1.161549e-08 5.184380e-08 2.917677e-08
## [22,] -4.658724e-07 3.991581e-04 -5.185087e-07 -5.173529e-08 6.884834e-09
## [23,] -1.161549e-08 -5.185087e-07 3.992760e-04 -2.299577e-07 -8.716798e-09
## [24,] 5.184380e-08 -5.173529e-08 -2.299577e-07 3.990245e-04 -3.078258e-07
## [25,] 2.917677e-08 6.884834e-09 -8.716798e-09 -3.078258e-07 3.997425e-04
## [26,] 6.352694e-09 1.830300e-08 3.928190e-08 1.677520e-08 -1.419510e-07
## [27,] -1.384945e-08 8.332051e-09 3.491080e-08 1.167103e-08 -1.480612e-08
## [28,] 6.549618e-08 3.038655e-08 4.013070e-08 8.436472e-08 4.339747e-08
## [29,] -2.400122e-08 -1.106234e-08 -5.213547e-09 -1.636407e-08 -5.742619e-09
## [30,] -3.282279e-07 -1.789428e-07 6.765372e-09 -8.902796e-09 -4.845001e-09
## [31,] -1.202379e-07 -4.874210e-08 4.575852e-08 3.249113e-08 1.475018e-08
## [32,] -3.683897e-08 -3.982595e-08 2.150809e-08 5.190066e-09 1.091179e-10
## [33,] 5.845736e-07 4.599435e-07 3.645337e-08 6.774783e-08 4.341583e-08
## [34,] 5.065132e-07 4.562761e-07 -1.120579e-07 2.261572e-08 3.881565e-08
## [35,] 1.881502e-08 -1.458803e-07 -3.561534e-07 -3.985045e-08 3.680409e-08
## [36,] -2.214067e-09 -4.215087e-08 -8.887679e-08 -1.446339e-07 3.848434e-08
## [37,] -6.260458e-09 1.409892e-09 6.118771e-09 2.365474e-08 4.910468e-08
## [38,] 3.472107e-09 2.650668e-08 7.549720e-08 5.668826e-08 -9.726716e-08
## [39,] 1.954188e-08 4.013262e-08 7.524979e-08 4.088640e-08 -2.883959e-09
## [40,] -8.727698e-09 -1.153946e-08 5.863491e-08 1.104750e-07 5.819638e-08
## [41,] 2.549935e-08 1.787629e-08 -2.904465e-09 -1.351930e-08 -3.522782e-09
## [42,] 1.023137e-06 5.601167e-07 1.128416e-07 2.286576e-07 1.246722e-07
## [43,] 2.591014e-07 2.760873e-07 1.571013e-07 -3.063758e-08 -4.515591e-08
## [44,] -1.138275e-06 5.632010e-08 7.764668e-08 7.237570e-08 3.454617e-08
## [45,] -3.807583e-06 -1.281143e-06 2.878320e-09 1.034829e-07 5.274413e-08
## [46,] -1.351409e-06 -1.952337e-06 -1.090920e-06 -1.025784e-07 1.006244e-08
## [47,] -1.587762e-08 -1.114289e-06 -1.411176e-06 -2.543826e-07 2.519011e-08
## [48,] 6.485841e-08 -1.073494e-07 -3.058899e-07 -1.451551e-06 -4.156326e-07
## [49,] 2.982450e-08 1.422796e-08 -4.392020e-09 -4.283067e-07 -2.531042e-07
## [50,] 1.313831e-08 5.928526e-08 1.557224e-07 1.079075e-07 -2.689377e-07
## [51,] -6.674219e-08 1.857110e-08 1.495601e-07 8.612595e-08 -6.935261e-09
## [52,] 2.462364e-07 1.312521e-07 1.250464e-07 2.254942e-07 1.152495e-07
## [53,] -1.092281e-07 -5.738779e-08 -1.602544e-08 -3.920547e-08 -1.161922e-08
## [54,] -2.596218e-06 -1.381205e-06 -6.243374e-08 -3.324771e-07 -1.895544e-07
## [55,] -6.546237e-07 -2.594248e-07 2.163309e-07 5.188859e-08 5.224053e-09
## [56,] 3.332664e-07 -1.207121e-07 7.874036e-08 3.409923e-08 1.058333e-08
## [57,] 3.222869e-06 1.863307e-06 8.073639e-08 3.376781e-07 2.228266e-07
## [58,] 2.161859e-06 1.537670e-06 -4.690461e-07 1.719163e-07 1.839569e-07
## [59,] 8.547662e-08 -5.457491e-07 -1.079644e-06 -9.732661e-08 1.035881e-07
## [60,] 4.712432e-08 -1.083866e-07 -2.511051e-07 -1.161144e-06 -2.143641e-07
## [61,] 1.956202e-08 1.261169e-08 1.395958e-08 -2.387025e-07 -1.273888e-07
## [62,] 1.758668e-08 1.143902e-07 3.162738e-07 2.277799e-07 -4.363286e-07
## [63,] 1.741404e-07 2.063853e-07 3.301654e-07 2.360302e-07 3.537076e-08
## [64,] -2.830047e-07 -1.538405e-07 1.829941e-07 2.776452e-07 1.375789e-07
## [65,] 2.265220e-07 1.261149e-07 -1.009417e-09 4.000372e-09 2.015995e-08
## [66,] 7.913297e-06 -1.554400e-06 9.980407e-07 1.880316e-05 1.202925e-05
## [,26] [,27] [,28] [,29] [,30]
## [1,] 2.304404e-07 1.415656e-07 -8.426327e-07 6.627573e-08 -2.061934e-08
## [2,] 3.658512e-08 7.250357e-08 -6.208568e-08 4.841503e-08 -6.363355e-07
## [3,] 3.793090e-08 4.695195e-08 -1.043983e-09 1.017957e-08 -1.688978e-07
## [4,] 3.331888e-08 4.325408e-08 -2.613577e-08 1.242478e-08 7.497963e-08
## [5,] 5.259152e-08 7.282597e-08 -5.767323e-08 2.508919e-08 2.556779e-07
## [6,] 3.556279e-08 4.843826e-08 -3.075804e-08 1.557847e-08 1.755251e-07
## [7,] 2.884986e-08 3.254177e-08 7.837583e-09 3.936364e-09 8.258053e-08
## [8,] 4.141242e-08 4.746746e-08 -4.677204e-08 1.674782e-08 1.043056e-07
## [9,] 6.360896e-08 3.420047e-08 -4.646579e-08 1.418938e-08 6.302207e-08
## [10,] -5.300243e-08 -6.824914e-08 -4.843254e-08 1.731460e-08 1.144535e-07
## [11,] -7.382049e-08 -2.594696e-07 -8.236903e-08 -3.671565e-08 2.172423e-07
## [12,] -1.417487e-09 -9.930537e-09 -2.451688e-07 1.560114e-09 -1.628440e-07
## [13,] 6.165407e-09 -5.158155e-08 -8.277902e-09 -2.166817e-07 1.372832e-07
## [14,] -9.836703e-07 -6.017211e-07 -1.196068e-06 2.076720e-07 1.847612e-07
## [15,] -6.433304e-07 -6.802591e-07 -7.068608e-07 -5.368592e-08 -8.153230e-07
## [16,] -2.971297e-06 -2.551129e-06 -2.918888e-06 5.769754e-07 5.807124e-06
## [17,] -2.719219e-06 -3.079801e-06 -7.076084e-07 -4.019227e-07 -1.158867e-05
## [18,] -1.354727e-08 9.326881e-08 -2.490361e-07 1.563958e-07 3.766824e-07
## [19,] 6.520777e-08 7.156066e-08 2.347962e-09 1.083509e-08 1.834484e-07
## [20,] 3.104110e-08 2.494567e-08 3.883211e-08 -7.431937e-09 -4.702023e-08
## [21,] 6.352694e-09 -1.384945e-08 6.549618e-08 -2.400122e-08 -3.282279e-07
## [22,] 1.830300e-08 8.332051e-09 3.038655e-08 -1.106234e-08 -1.789428e-07
## [23,] 3.928190e-08 3.491080e-08 4.013070e-08 -5.213547e-09 6.765372e-09
## [24,] 1.677520e-08 1.167103e-08 8.436472e-08 -1.636407e-08 -8.902796e-09
## [25,] -1.419510e-07 -1.480612e-08 4.339747e-08 -5.742619e-09 -4.845001e-09
## [26,] 3.993383e-04 -3.242785e-07 -8.870097e-08 4.689661e-08 1.771978e-07
## [27,] -3.242785e-07 3.993818e-04 -7.428807e-08 -1.265505e-07 2.347596e-07
## [28,] -8.870097e-08 -7.428807e-08 3.995016e-04 9.079729e-08 -2.015001e-08
## [29,] 4.689661e-08 -1.265505e-07 9.079729e-08 3.996655e-04 6.974033e-08
## [30,] 1.771978e-07 2.347596e-07 -2.015001e-08 6.974033e-08 3.976545e-04
## [31,] 1.671816e-07 1.755682e-07 1.015006e-07 5.084521e-09 -5.532032e-07
## [32,] 8.098565e-08 9.133988e-08 2.963941e-08 1.148777e-08 2.005857e-07
## [33,] 1.007527e-07 1.198156e-07 7.565070e-08 1.824869e-08 7.847681e-07
## [34,] 6.509734e-08 7.418824e-08 6.776052e-08 7.175647e-09 4.710374e-07
## [35,] 5.812065e-08 5.359949e-08 8.828622e-08 -8.039615e-09 1.585255e-07
## [36,] 5.739879e-08 5.202105e-08 5.368752e-08 -2.412441e-09 1.243155e-07
## [37,] -9.027286e-08 1.306455e-08 1.055511e-08 6.768411e-09 7.096106e-08
## [38,] -8.561429e-07 -4.296435e-07 -9.723588e-08 6.823894e-08 3.757705e-07
## [39,] -4.398156e-07 -9.274066e-07 -2.361846e-07 -1.602376e-07 6.292289e-07
## [40,] -7.981326e-08 -2.111992e-07 -4.842609e-07 1.728804e-08 -3.858495e-07
## [41,] 4.646223e-08 -1.846222e-07 8.150131e-09 -3.503099e-07 3.380235e-07
## [42,] -2.419588e-07 1.543149e-08 -7.048319e-07 3.963935e-07 4.582385e-06
## [43,] 2.609097e-07 2.994428e-07 -1.718146e-08 6.122607e-08 1.269070e-06
## [44,] 1.035545e-07 8.526061e-08 1.191392e-07 -2.234553e-08 -2.247884e-07
## [45,] -1.709763e-08 -8.344514e-08 1.507858e-07 -7.714988e-08 -1.349838e-06
## [46,] 3.144548e-08 7.242513e-11 7.236154e-08 -3.508455e-08 -6.659833e-07
## [47,] 9.845610e-08 8.364031e-08 1.359310e-07 -1.944967e-08 4.441647e-08
## [48,] 8.031954e-08 6.638218e-08 1.411305e-07 -2.061334e-08 8.419529e-08
## [49,] -2.727206e-07 -3.699919e-09 5.753705e-08 2.475885e-09 6.609438e-08
## [50,] -1.944098e-06 -9.646572e-07 -2.224654e-07 1.599995e-07 7.576371e-07
## [51,] -9.429497e-07 -1.840622e-06 -3.924879e-07 -3.271565e-07 6.853742e-07
## [52,] -3.023435e-07 -4.928702e-07 -1.086537e-06 1.900719e-07 6.321451e-07
## [53,] 1.665569e-07 -3.216630e-07 1.954679e-07 -5.902105e-07 -1.602659e-07
## [54,] 1.151155e-06 1.101726e-06 7.519719e-07 -1.573042e-07 -1.500217e-05
## [55,] 8.683969e-07 8.638870e-07 5.588906e-07 -3.994178e-08 -3.334644e-06
## [56,] 2.970636e-07 3.337105e-07 1.151535e-07 4.060101e-08 7.297109e-07
## [57,] 3.226419e-07 3.894039e-07 3.152132e-07 6.078775e-08 3.065684e-06
## [58,] 2.004861e-07 2.248863e-07 2.740403e-07 1.530521e-08 1.684969e-06
## [59,] 1.672219e-07 1.500234e-07 2.677838e-07 -2.796077e-08 4.217451e-07
## [60,] 1.460341e-07 1.257446e-07 2.056288e-07 -2.246610e-08 2.706689e-07
## [61,] -4.546384e-07 6.225918e-09 7.687360e-08 1.336720e-08 1.796828e-07
## [62,] -3.638440e-06 -1.855416e-06 -4.105649e-07 3.240192e-07 1.572070e-06
## [63,] -1.908258e-06 -3.518740e-06 -1.078868e-06 -4.182323e-07 2.802106e-06
## [64,] -3.526062e-07 -1.014410e-06 -1.219132e-06 1.315142e-08 -2.062801e-06
## [65,] 1.772239e-07 -5.788089e-07 -3.265516e-08 -7.311689e-07 1.615359e-06
## [66,] -3.571892e-06 -6.210078e-06 1.501497e-05 -3.646624e-06 -8.113671e-06
## [,31] [,32] [,33] [,34] [,35]
## [1,] -6.933501e-07 -6.555488e-07 -1.679837e-06 -1.648317e-06 -1.488793e-06
## [2,] -2.509352e-07 1.370992e-08 1.197064e-07 7.301378e-08 2.991577e-08
## [3,] -9.713017e-07 -2.455097e-07 7.258005e-08 5.208042e-08 2.125771e-08
## [4,] -2.224666e-07 -4.315758e-07 -4.056090e-07 -1.050214e-08 -1.252219e-08
## [5,] 1.283729e-07 -3.437790e-07 -1.264772e-06 -4.871504e-07 -5.861372e-08
## [6,] 9.982544e-08 4.574408e-08 -4.400080e-07 -8.385078e-07 -5.306391e-07
## [7,] 6.480777e-08 3.700744e-08 2.055843e-08 -4.893843e-07 -7.171173e-07
## [8,] 7.048046e-08 4.909661e-08 -8.151899e-09 -8.283728e-08 -2.813233e-07
## [9,] 3.882008e-08 3.103637e-08 -3.028220e-08 -3.214399e-08 -5.765006e-08
## [10,] 9.724019e-08 5.464245e-08 3.104888e-08 1.347222e-08 6.762329e-09
## [11,] 9.168151e-08 5.259082e-08 -1.078186e-08 -2.093903e-08 -2.034931e-08
## [12,] 8.581286e-08 5.002684e-08 7.232058e-08 4.962202e-08 3.900340e-08
## [13,] -8.493245e-09 -6.532784e-09 -2.455242e-08 -1.812173e-08 -1.211695e-08
## [14,] 3.318583e-07 -1.234095e-07 1.134532e-06 9.066528e-07 -2.169467e-07
## [15,] -2.023100e-06 -1.640200e-06 -2.840231e-06 -2.253625e-06 -1.525419e-06
## [16,] -1.547125e-06 -7.231620e-07 4.482046e-06 2.445425e-06 -1.173710e-06
## [17,] -6.916259e-06 -3.180963e-06 -6.555536e-06 -3.870423e-06 -1.618253e-06
## [18,] -1.885821e-08 -1.040821e-07 -2.196557e-07 -1.209271e-07 2.812004e-09
## [19,] 3.012817e-08 -5.148456e-08 -1.092092e-07 -7.155106e-08 -6.499181e-09
## [20,] -5.025285e-08 -1.052054e-07 5.697047e-09 -1.905936e-08 1.996509e-08
## [21,] -1.202379e-07 -3.683897e-08 5.845736e-07 5.065132e-07 1.881502e-08
## [22,] -4.874210e-08 -3.982595e-08 4.599435e-07 4.562761e-07 -1.458803e-07
## [23,] 4.575852e-08 2.150809e-08 3.645337e-08 -1.120579e-07 -3.561534e-07
## [24,] 3.249113e-08 5.190066e-09 6.774783e-08 2.261572e-08 -3.985045e-08
## [25,] 1.475018e-08 1.091179e-10 4.341583e-08 3.881565e-08 3.680409e-08
## [26,] 1.671816e-07 8.098565e-08 1.007527e-07 6.509734e-08 5.812065e-08
## [27,] 1.755682e-07 9.133988e-08 1.198156e-07 7.418824e-08 5.359949e-08
## [28,] 1.015006e-07 2.963941e-08 7.565070e-08 6.776052e-08 8.828622e-08
## [29,] 5.084521e-09 1.148777e-08 1.824869e-08 7.175647e-09 -8.039615e-09
## [30,] -5.532032e-07 2.005857e-07 7.847681e-07 4.710374e-07 1.585255e-07
## [31,] 3.970446e-04 -6.272863e-07 4.071508e-07 2.652576e-07 1.398001e-07
## [32,] -6.272863e-07 3.989712e-04 -9.870679e-07 9.488640e-08 6.482000e-08
## [33,] 4.071508e-07 -9.870679e-07 3.966245e-04 -1.094423e-06 1.252363e-07
## [34,] 2.652576e-07 9.488640e-08 -1.094423e-06 3.982037e-04 -9.737865e-07
## [35,] 1.398001e-07 6.482000e-08 1.252363e-07 -9.737865e-07 3.986557e-04
## [36,] 1.148303e-07 5.539028e-08 1.150922e-07 -4.379936e-08 -2.885497e-07
## [37,] 6.175361e-08 3.158244e-08 4.435937e-08 3.084388e-08 -1.026496e-08
## [38,] 3.466484e-07 1.674063e-07 2.328815e-07 1.535253e-07 1.281391e-07
## [39,] 3.298837e-07 1.473197e-07 1.710071e-07 1.116314e-07 1.072839e-07
## [40,] 2.700970e-07 1.530567e-07 3.413789e-07 2.380978e-07 1.728419e-07
## [41,] -3.977533e-08 -3.605762e-08 -9.243255e-08 -6.048258e-08 -2.913078e-08
## [42,] -7.090517e-07 -9.415184e-07 -1.769404e-06 -1.019898e-06 -1.537320e-07
## [43,] -1.576014e-06 -6.200431e-07 -4.579247e-07 -3.201929e-07 -2.377332e-08
## [44,] -6.142248e-07 -7.225792e-08 4.341173e-07 -6.766188e-08 5.606348e-08
## [45,] -5.363055e-07 2.347521e-07 3.007323e-06 1.953928e-06 -6.313507e-08
## [46,] -2.173945e-07 -1.524919e-07 1.948921e-06 1.543758e-06 -6.046389e-07
## [47,] 1.362282e-07 5.694882e-08 1.907727e-07 -3.971395e-07 -1.049976e-06
## [48,] 1.363264e-07 5.398948e-08 1.514908e-07 1.082968e-08 -1.942263e-07
## [49,] 8.384394e-08 3.471451e-08 8.396524e-08 6.709524e-08 3.595323e-08
## [50,] 7.012991e-07 3.371883e-07 4.609375e-07 3.023660e-07 2.562230e-07
## [51,] 7.560210e-07 4.054438e-07 5.981705e-07 3.799469e-07 2.675851e-07
## [52,] 3.214569e-07 3.996381e-08 6.978354e-08 1.003813e-07 2.229294e-07
## [53,] 3.160191e-08 7.582209e-08 1.577205e-07 8.606206e-08 2.513920e-10
## [54,] -1.211958e-06 1.978365e-06 5.434308e-06 3.188967e-06 8.548339e-07
## [55,] -1.108035e-05 -1.970231e-06 2.077507e-06 1.261034e-06 5.989055e-07
## [56,] -2.367191e-06 -3.129192e-06 -3.194215e-06 3.500857e-07 2.420886e-07
## [57,] 1.559209e-06 -3.154680e-06 -1.199533e-05 -3.934757e-06 6.370627e-07
## [58,] 9.157635e-07 3.298074e-07 -4.119222e-06 -5.315420e-06 -2.250528e-06
## [59,] 3.867483e-07 1.737232e-07 3.339195e-07 -2.426388e-06 -3.275202e-06
## [60,] 2.872345e-07 1.255851e-07 3.289519e-07 -6.662100e-08 -5.378339e-07
## [61,] 1.791907e-07 8.185267e-08 1.659190e-07 1.246225e-07 3.319196e-08
## [62,] 1.445762e-06 6.949025e-07 9.620267e-07 6.312407e-07 5.282328e-07
## [63,] 1.372137e-06 5.560588e-07 6.436762e-07 4.401327e-07 4.794553e-07
## [64,] 1.076204e-06 7.440060e-07 1.534086e-06 1.004426e-06 5.893129e-07
## [65,] -2.147880e-07 -2.481778e-07 -5.314212e-07 -3.222827e-07 -1.062475e-07
## [66,] -3.140865e-06 -4.359057e-06 1.756284e-05 1.582217e-05 1.449993e-05
## [,36] [,37] [,38] [,39] [,40]
## [1,] -1.485382e-06 -6.110354e-07 -4.960866e-07 -6.117359e-08 -1.422390e-06
## [2,] 2.932432e-08 1.763931e-08 7.916099e-08 1.846734e-07 -2.114870e-07
## [3,] 2.714374e-08 1.867346e-08 8.089889e-08 7.660015e-08 4.939768e-08
## [4,] 1.059529e-08 1.694071e-08 6.496812e-08 6.633759e-08 5.758143e-09
## [5,] 1.462491e-08 2.823296e-08 1.036155e-07 1.045342e-07 4.609202e-09
## [6,] -4.935546e-08 2.034854e-08 7.236140e-08 7.054240e-08 1.611696e-08
## [7,] -2.292959e-07 -6.139984e-09 6.052649e-08 5.701671e-08 3.950314e-08
## [8,] -1.044457e-06 -3.355641e-07 5.925803e-08 7.268306e-08 -2.460865e-08
## [9,] -3.581262e-07 -2.736634e-07 -1.117344e-07 2.981553e-08 -4.478314e-08
## [10,] 1.729536e-08 -1.291621e-07 -6.461731e-07 -3.099615e-07 -1.050269e-07
## [11,] 1.180049e-08 4.704707e-09 -3.130160e-07 -6.021475e-07 -1.179738e-07
## [12,] 3.562465e-08 1.416302e-08 -4.771858e-08 -3.207234e-08 -5.097186e-07
## [13,] -6.956096e-09 2.324672e-09 3.824583e-08 -1.325381e-07 8.261071e-08
## [14,] -2.395415e-08 2.813712e-08 -1.063662e-06 -1.152495e-06 -9.642735e-07
## [15,] -1.467374e-06 -6.800692e-07 -1.577413e-06 -1.307782e-06 -1.631076e-06
## [16,] -1.347727e-06 -8.007801e-07 -5.711723e-06 -6.247480e-06 -2.684738e-06
## [17,] -1.389005e-06 -8.778874e-07 -5.475671e-06 -4.990344e-06 -4.528877e-06
## [18,] -4.543815e-09 -1.090199e-08 -3.321743e-08 1.621324e-08 -1.102434e-07
## [19,] 1.424805e-08 1.752179e-08 1.166221e-07 1.208133e-07 4.853672e-08
## [20,] 1.656867e-08 7.466427e-09 5.845881e-08 6.103419e-08 4.134035e-08
## [21,] -2.214067e-09 -6.260458e-09 3.472107e-09 1.954188e-08 -8.727698e-09
## [22,] -4.215087e-08 1.409892e-09 2.650668e-08 4.013262e-08 -1.153946e-08
## [23,] -8.887679e-08 6.118771e-09 7.549720e-08 7.524979e-08 5.863491e-08
## [24,] -1.446339e-07 2.365474e-08 5.668826e-08 4.088640e-08 1.104750e-07
## [25,] 3.848434e-08 4.910468e-08 -9.726716e-08 -2.883959e-09 5.819638e-08
## [26,] 5.739879e-08 -9.027286e-08 -8.561429e-07 -4.398156e-07 -7.981326e-08
## [27,] 5.202105e-08 1.306455e-08 -4.296435e-07 -9.274066e-07 -2.111992e-07
## [28,] 5.368752e-08 1.055511e-08 -9.723588e-08 -2.361846e-07 -4.842609e-07
## [29,] -2.412441e-09 6.768411e-09 6.823894e-08 -1.602376e-07 1.728804e-08
## [30,] 1.243155e-07 7.096106e-08 3.757705e-07 6.292289e-07 -3.858495e-07
## [31,] 1.148303e-07 6.175361e-08 3.466484e-07 3.298837e-07 2.700970e-07
## [32,] 5.539028e-08 3.158244e-08 1.674063e-07 1.473197e-07 1.530567e-07
## [33,] 1.150922e-07 4.435937e-08 2.328815e-07 1.710071e-07 3.413789e-07
## [34,] -4.379936e-08 3.084388e-08 1.535253e-07 1.116314e-07 2.380978e-07
## [35,] -2.885497e-07 -1.026496e-08 1.281391e-07 1.072839e-07 1.728419e-07
## [36,] 3.984860e-04 -4.614514e-07 1.120362e-07 1.004800e-07 1.165032e-07
## [37,] -4.614514e-07 3.997274e-04 -2.555339e-07 1.573806e-08 3.463997e-08
## [38,] 1.120362e-07 -2.555339e-07 3.980494e-04 -9.956170e-07 -1.781836e-07
## [39,] 1.004800e-07 1.573806e-08 -9.956170e-07 3.980848e-04 -3.816855e-07
## [40,] 1.165032e-07 3.463997e-08 -1.781836e-07 -3.816855e-07 3.989888e-04
## [41,] -1.844407e-08 3.940241e-09 1.124514e-07 -3.711052e-07 1.588412e-07
## [42,] -1.432792e-07 -1.162216e-07 -5.269711e-07 -6.223709e-07 3.212853e-08
## [43,] 6.420898e-08 7.355391e-08 4.709235e-07 4.284996e-07 3.400830e-07
## [44,] 5.082140e-08 2.481586e-08 1.944355e-07 2.029401e-07 1.316506e-07
## [45,] -6.091056e-08 -3.297156e-08 -7.613619e-08 4.790520e-09 -1.953126e-07
## [46,] -1.436087e-07 -2.748249e-09 3.167120e-08 8.190687e-08 -9.748940e-08
## [47,] -2.511895e-07 -3.018218e-09 1.972730e-07 1.864481e-07 2.038998e-07
## [48,] -1.243673e-06 -2.844281e-07 1.751465e-07 1.499137e-07 2.206786e-07
## [49,] -2.656168e-07 -1.509667e-07 -4.434107e-07 1.156090e-08 9.685374e-08
## [50,] 2.318706e-07 -4.118071e-07 -3.663991e-06 -1.929661e-06 -3.285823e-07
## [51,] 2.381560e-07 6.438270e-08 -1.837429e-06 -3.495043e-06 -9.530332e-07
## [52,] 1.426896e-07 2.571688e-08 -5.781930e-07 -1.266542e-06 -1.309987e-06
## [53,] 6.713115e-09 2.994392e-08 3.385397e-07 -4.089774e-07 3.493619e-08
## [54,] 7.228871e-07 4.696554e-07 2.425151e-06 3.602710e-06 -1.548019e-06
## [55,] 5.291676e-07 3.142671e-07 1.772533e-06 1.808420e-06 1.009371e-06
## [56,] 2.000209e-07 1.119440e-07 6.125177e-07 5.355741e-07 5.770584e-07
## [57,] 4.391676e-07 1.425603e-07 7.785550e-07 5.182179e-07 1.350364e-06
## [58,] -1.939050e-08 8.981100e-08 4.882032e-07 3.265015e-07 8.863995e-07
## [59,] -5.721107e-07 -1.566859e-08 3.651781e-07 3.065681e-07 5.034411e-07
## [60,] -3.323526e-06 -9.823947e-07 3.054907e-07 2.611856e-07 3.724199e-07
## [61,] -9.543276e-07 -5.712252e-07 -8.968506e-07 1.927105e-08 1.550261e-07
## [62,] 4.706629e-07 -8.098139e-07 -7.530731e-06 -4.122257e-06 -7.109106e-07
## [63,] 4.262831e-07 1.019899e-07 -4.139729e-06 -7.424245e-06 -1.736432e-06
## [64,] 4.235059e-07 1.614874e-07 -7.535748e-07 -1.807791e-06 -2.889274e-06
## [65,] -8.135638e-08 -8.715509e-09 3.990821e-07 -1.170787e-06 3.912310e-07
## [66,] 4.451148e-06 -2.014438e-06 -4.765784e-06 -8.077459e-06 1.737217e-05
## [,41] [,42] [,43] [,44] [,45]
## [1,] 2.121388e-07 -3.385292e-06 1.595862e-06 -6.455058e-08 -3.812199e-08
## [2,] 1.470179e-07 2.224860e-07 2.573059e-07 -3.008161e-09 -1.882367e-07
## [3,] -9.387142e-09 -3.589126e-07 6.559765e-08 -5.889163e-08 -1.553674e-07
## [4,] -3.158845e-09 -3.513635e-07 8.918269e-08 -1.107162e-07 -9.357741e-08
## [5,] -8.085157e-09 -7.246068e-07 1.811941e-07 -6.295850e-09 4.525079e-07
## [6,] -7.937082e-09 -4.828625e-07 9.461212e-08 -3.095669e-08 3.854305e-07
## [7,] -6.723492e-09 -1.604329e-07 7.349475e-08 1.471234e-08 -6.169216e-08
## [8,] 3.536067e-09 -3.245144e-07 2.115852e-07 -3.661455e-09 -1.074642e-07
## [9,] 1.028254e-08 -2.370381e-07 1.750903e-07 -8.066384e-09 -6.921501e-08
## [10,] 4.031094e-08 -2.383833e-07 2.093632e-07 3.964872e-08 -5.029490e-08
## [11,] -1.281470e-07 -2.170094e-07 2.770262e-07 3.814294e-08 -4.112565e-08
## [12,] 8.836376e-08 -2.793204e-07 1.252991e-07 4.644841e-08 -3.526280e-08
## [13,] -3.316838e-07 1.046124e-07 1.281636e-08 -1.531032e-09 1.801610e-08
## [14,] -9.967221e-08 -3.254780e-06 -4.418504e-06 -2.442520e-06 -4.938748e-06
## [15,] 2.000906e-07 -1.536366e-06 -5.735761e-07 1.358640e-07 2.612220e-06
## [16,] -4.859037e-07 -2.537992e-05 -1.991351e-05 -8.153056e-06 -1.634376e-05
## [17,] 1.067277e-06 2.662227e-05 -1.894762e-06 -3.413739e-07 9.751620e-06
## [18,] 7.642330e-08 -2.794083e-06 -1.453744e-06 -6.292478e-08 5.714363e-07
## [19,] -2.909645e-09 -1.363641e-06 -3.391793e-06 -7.975880e-07 2.223115e-07
## [20,] 1.784325e-09 4.344068e-08 -7.541425e-07 -1.110424e-06 -1.138912e-06
## [21,] 2.549935e-08 1.023137e-06 2.591014e-07 -1.138275e-06 -3.807583e-06
## [22,] 1.787629e-08 5.601167e-07 2.760873e-07 5.632010e-08 -1.281143e-06
## [23,] -2.904465e-09 1.128416e-07 1.571013e-07 7.764668e-08 2.878320e-09
## [24,] -1.351930e-08 2.286576e-07 -3.063758e-08 7.237570e-08 1.034829e-07
## [25,] -3.522782e-09 1.246722e-07 -4.515591e-08 3.454617e-08 5.274413e-08
## [26,] 4.646223e-08 -2.419588e-07 2.609097e-07 1.035545e-07 -1.709763e-08
## [27,] -1.846222e-07 1.543149e-08 2.994428e-07 8.526061e-08 -8.344514e-08
## [28,] 8.150131e-09 -7.048319e-07 -1.718146e-08 1.191392e-07 1.507858e-07
## [29,] -3.503099e-07 3.963935e-07 6.122607e-08 -2.234553e-08 -7.714988e-08
## [30,] 3.380235e-07 4.582385e-06 1.269070e-06 -2.247884e-07 -1.349838e-06
## [31,] -3.977533e-08 -7.090517e-07 -1.576014e-06 -6.142248e-07 -5.363055e-07
## [32,] -3.605762e-08 -9.415184e-07 -6.200431e-07 -7.225792e-08 2.347521e-07
## [33,] -9.243255e-08 -1.769404e-06 -4.579247e-07 4.341173e-07 3.007323e-06
## [34,] -6.048258e-08 -1.019898e-06 -3.201929e-07 -6.766188e-08 1.953928e-06
## [35,] -2.913078e-08 -1.537320e-07 -2.377332e-08 5.606348e-08 -6.313507e-08
## [36,] -1.844407e-08 -1.432792e-07 6.420898e-08 5.082140e-08 -6.091056e-08
## [37,] 3.940241e-09 -1.162216e-07 7.355391e-08 2.481586e-08 -3.297156e-08
## [38,] 1.124514e-07 -5.269711e-07 4.709235e-07 1.944355e-07 -7.613619e-08
## [39,] -3.711052e-07 -6.223709e-07 4.284996e-07 2.029401e-07 4.790520e-09
## [40,] 1.588412e-07 3.212853e-08 3.400830e-07 1.316506e-07 -1.953126e-07
## [41,] 3.994187e-04 -8.514268e-08 -9.344919e-08 4.612755e-09 1.232218e-07
## [42,] -8.514268e-08 3.822979e-04 -7.591749e-06 -7.820667e-08 3.891695e-06
## [43,] -9.344919e-08 -7.591749e-06 3.864774e-04 -2.904558e-06 1.101948e-06
## [44,] 4.612755e-09 -7.820667e-08 -2.904558e-06 3.965626e-04 -3.743839e-06
## [45,] 1.232218e-07 3.891695e-06 1.101948e-06 -3.743839e-06 3.862803e-04
## [46,] 6.864754e-08 1.961088e-06 8.791247e-07 1.454904e-07 -4.965934e-06
## [47,] -1.899201e-08 2.886520e-07 2.722880e-07 1.930836e-07 -8.677074e-08
## [48,] -2.800075e-08 1.733149e-07 9.412374e-08 1.463500e-07 1.055324e-07
## [49,] 4.674788e-09 2.804658e-08 6.547894e-08 7.492268e-08 4.260763e-08
## [50,] 2.274672e-07 -1.056357e-06 9.808965e-07 4.046645e-07 -1.325754e-07
## [51,] -5.259622e-07 2.443654e-07 1.277702e-06 3.583863e-07 -4.316950e-07
## [52,] -1.244180e-08 -3.558722e-06 -3.149636e-07 3.969837e-07 6.922408e-07
## [53,] -7.354975e-07 1.946680e-06 3.815724e-07 -8.374967e-08 -3.968751e-07
## [54,] 1.501045e-06 4.042431e-05 9.832291e-06 -1.926610e-06 -1.028764e-05
## [55,] -4.665577e-08 6.822682e-07 -1.042267e-05 -3.787341e-06 -2.828617e-06
## [56,] -1.380047e-07 -3.923758e-06 -3.969371e-06 4.333893e-07 2.149683e-06
## [57,] -3.780203e-07 -6.986697e-06 -2.287844e-06 2.666710e-06 1.511125e-05
## [58,] -2.287294e-07 -3.635719e-06 -1.443921e-06 -2.025274e-07 8.682975e-06
## [59,] -8.278386e-08 -3.320289e-07 -6.277747e-08 1.835398e-07 -4.109897e-08
## [60,] -5.567269e-08 -9.991132e-08 9.280818e-08 1.831940e-07 -3.930105e-08
## [61,] 1.122520e-08 -1.553443e-07 1.426685e-07 1.083333e-07 -1.968500e-08
## [62,] 5.095831e-07 -2.218988e-06 1.982834e-06 8.183503e-07 -3.068736e-07
## [63,] -1.052408e-06 -4.323876e-06 1.346055e-06 9.164065e-07 3.164284e-07
## [64,] 4.231532e-07 4.207943e-06 2.410581e-06 3.256870e-07 -1.576356e-06
## [65,] -1.454784e-06 -2.775501e-06 -1.030463e-06 8.559059e-08 9.297150e-07
## [66,] -3.046768e-06 5.024874e-05 -4.958148e-05 2.047514e-06 9.200804e-06
## [,46] [,47] [,48] [,49] [,50]
## [1,] 5.648680e-07 -5.661182e-07 -4.664659e-07 -9.888857e-08 -7.405878e-08
## [2,] -8.474648e-08 1.430628e-08 1.326195e-08 1.076593e-08 1.558495e-07
## [3,] -5.261757e-08 1.169984e-08 3.605053e-09 6.293541e-09 1.591564e-07
## [4,] -3.402557e-08 -1.762835e-08 -3.950586e-08 -1.424979e-08 1.286055e-07
## [5,] 5.002210e-07 -1.592823e-08 -8.093377e-08 -3.045218e-08 2.046428e-07
## [6,] 4.609527e-07 -1.399748e-07 -8.000707e-08 -1.440266e-08 1.418476e-07
## [7,] -1.599959e-07 -3.626144e-07 -9.988733e-08 2.729362e-09 1.190451e-07
## [8,] -4.379038e-08 -1.512954e-07 -2.459030e-07 -6.946297e-09 1.340431e-07
## [9,] 7.614437e-09 -5.047201e-08 -5.082716e-08 1.894328e-08 -4.289829e-08
## [10,] 1.453747e-08 1.703735e-08 -1.872062e-09 -1.179482e-07 -8.234870e-07
## [11,] 3.725663e-08 -5.636272e-09 -4.042478e-08 -3.165773e-08 -4.032132e-07
## [12,] -2.149155e-09 4.927078e-08 4.159548e-08 1.875572e-08 -2.051708e-08
## [13,] 1.422857e-08 -9.861763e-09 -1.364624e-08 -1.443093e-09 5.187424e-08
## [14,] -3.717637e-06 -2.141211e-06 -1.685513e-06 -7.352686e-07 -2.509340e-06
## [15,] 1.498570e-06 -5.133265e-07 -5.007667e-07 -2.441193e-07 -2.716963e-06
## [16,] -1.021831e-05 -4.771502e-06 -3.286957e-06 -1.770091e-06 -1.182860e-05
## [17,] 4.035352e-06 -1.613704e-06 -1.561851e-06 -1.095286e-06 -1.125308e-05
## [18,] 2.899926e-07 7.871412e-08 6.293847e-08 2.352448e-08 -6.451773e-08
## [19,] 1.982306e-07 6.406386e-08 2.208698e-08 1.593752e-08 2.440870e-07
## [20,] 3.952873e-08 6.050796e-08 4.762152e-08 2.404978e-08 1.214071e-07
## [21,] -1.351409e-06 -1.587762e-08 6.485841e-08 2.982450e-08 1.313831e-08
## [22,] -1.952337e-06 -1.114289e-06 -1.073494e-07 1.422796e-08 5.928526e-08
## [23,] -1.090920e-06 -1.411176e-06 -3.058899e-07 -4.392020e-09 1.557224e-07
## [24,] -1.025784e-07 -2.543826e-07 -1.451551e-06 -4.283067e-07 1.079075e-07
## [25,] 1.006244e-08 2.519011e-08 -4.156326e-07 -2.531042e-07 -2.689377e-07
## [26,] 3.144548e-08 9.845610e-08 8.031954e-08 -2.727206e-07 -1.944098e-06
## [27,] 7.242513e-11 8.364031e-08 6.638218e-08 -3.699919e-09 -9.646572e-07
## [28,] 7.236154e-08 1.359310e-07 1.411305e-07 5.753705e-08 -2.224654e-07
## [29,] -3.508455e-08 -1.944967e-08 -2.061334e-08 2.475885e-09 1.599995e-07
## [30,] -6.659833e-07 4.441647e-08 8.419529e-08 6.609438e-08 7.576371e-07
## [31,] -2.173945e-07 1.362282e-07 1.363264e-07 8.384394e-08 7.012991e-07
## [32,] -1.524919e-07 5.694882e-08 5.398948e-08 3.471451e-08 3.371883e-07
## [33,] 1.948921e-06 1.907727e-07 1.514908e-07 8.396524e-08 4.609375e-07
## [34,] 1.543758e-06 -3.971395e-07 1.082968e-08 6.709524e-08 3.023660e-07
## [35,] -6.046389e-07 -1.049976e-06 -1.942263e-07 3.595323e-08 2.562230e-07
## [36,] -1.436087e-07 -2.511895e-07 -1.243673e-06 -2.656168e-07 2.318706e-07
## [37,] -2.748249e-09 -3.018218e-09 -2.844281e-07 -1.509667e-07 -4.118071e-07
## [38,] 3.167120e-08 1.972730e-07 1.751465e-07 -4.434107e-07 -3.663991e-06
## [39,] 8.190687e-08 1.864481e-07 1.499137e-07 1.156090e-08 -1.929661e-06
## [40,] -9.748940e-08 2.038998e-07 2.206786e-07 9.685374e-08 -3.285823e-07
## [41,] 6.864754e-08 -1.899201e-08 -2.800075e-08 4.674788e-09 2.274672e-07
## [42,] 1.961088e-06 2.886520e-07 1.733149e-07 2.804658e-08 -1.056357e-06
## [43,] 8.791247e-07 2.722880e-07 9.412374e-08 6.547894e-08 9.808965e-07
## [44,] 1.454904e-07 1.930836e-07 1.463500e-07 7.492268e-08 4.046645e-07
## [45,] -4.965934e-06 -8.677074e-08 1.055324e-07 4.260763e-08 -1.325754e-07
## [46,] 3.940255e-04 -2.834027e-06 -2.874749e-07 2.394622e-08 7.993605e-08
## [47,] -2.834027e-06 3.965426e-04 -5.811389e-07 2.193741e-08 4.052993e-07
## [48,] -2.874749e-07 -5.811389e-07 3.967173e-04 -9.360323e-07 3.523305e-07
## [49,] 2.394622e-08 2.193741e-08 -9.360323e-07 3.994461e-04 -8.769957e-07
## [50,] 7.993605e-08 4.052993e-07 3.523305e-07 -8.769957e-07 3.924452e-04
## [51,] -6.344467e-08 3.857872e-07 3.312849e-07 5.488035e-08 -3.956297e-06
## [52,] 3.655148e-07 3.960385e-07 3.858658e-07 1.529807e-07 -1.235089e-06
## [53,] -1.958011e-07 -4.907719e-08 -4.168412e-08 2.391555e-08 7.371284e-07
## [54,] -5.037124e-06 -8.671168e-08 1.721916e-07 2.697005e-07 4.898999e-06
## [55,] -1.140181e-06 5.756856e-07 5.398736e-07 3.607704e-07 3.601265e-06
## [56,] -4.517727e-07 2.184847e-07 2.134939e-07 1.350074e-07 1.237830e-06
## [57,] 8.284322e-06 6.828654e-07 6.417429e-07 3.451225e-07 1.536720e-06
## [58,] 5.936469e-06 -1.499984e-06 1.371419e-07 2.638056e-07 9.634531e-07
## [59,] -2.129818e-06 -3.469751e-06 -5.902032e-07 9.950698e-08 7.354653e-07
## [60,] -4.150470e-07 -7.196111e-07 -4.237989e-06 -1.053039e-06 6.202978e-07
## [61,] 6.332221e-09 4.028426e-08 -1.059868e-06 -7.061983e-07 -1.654830e-06
## [62,] 1.420608e-07 8.261575e-07 7.239407e-07 -1.607439e-06 -1.515291e-05
## [63,] 4.685126e-07 8.487383e-07 7.039471e-07 1.452341e-07 -8.703126e-06
## [64,] -7.379425e-07 6.156644e-07 6.697034e-07 3.136289e-07 -1.618974e-06
## [65,] 4.660659e-07 -2.450515e-08 -5.605253e-08 2.721466e-08 8.884749e-07
## [66,] -7.464115e-06 1.460847e-05 2.248193e-05 9.805632e-06 -9.925606e-06
## [,51] [,52] [,53] [,54] [,55]
## [1,] 2.067333e-07 -2.037791e-06 3.406248e-07 6.652979e-06 5.872295e-07
## [2,] 2.227522e-07 -7.381924e-08 8.501150e-08 -2.006045e-06 -6.770527e-07
## [3,] 1.948957e-07 -9.530580e-09 4.121024e-08 3.072726e-07 -2.748992e-06
## [4,] 1.625305e-07 -7.110718e-08 4.217995e-08 8.613631e-07 -3.448906e-07
## [5,] 2.737586e-07 -1.640797e-07 8.734880e-08 2.060142e-06 8.681840e-07
## [6,] 1.871179e-07 -9.120175e-08 5.656584e-08 1.382170e-06 5.904268e-07
## [7,] 1.351514e-07 2.134079e-08 1.691555e-08 5.706166e-07 3.370151e-07
## [8,] 1.639790e-07 -1.126813e-07 4.593442e-08 9.097605e-07 4.634472e-07
## [9,] 8.418713e-08 -1.189492e-07 4.079599e-08 6.159209e-07 2.965651e-07
## [10,] -3.737835e-07 -2.045003e-07 9.049473e-08 8.507702e-07 5.510101e-07
## [11,] -8.617700e-07 -3.840319e-07 -1.328485e-07 1.154371e-06 5.753613e-07
## [12,] -1.276670e-07 -5.533531e-07 4.021466e-08 1.662452e-08 4.309812e-07
## [13,] -1.784669e-07 1.191914e-08 -3.522331e-07 2.677945e-07 -3.116697e-08
## [14,] -2.034879e-06 -2.644848e-06 4.757786e-07 3.137190e-06 -2.836289e-07
## [15,] -2.655874e-06 -2.159946e-06 2.323134e-08 -4.061235e-07 -6.546366e-06
## [16,] -1.030370e-05 -1.119342e-05 2.761807e-06 5.044270e-05 -6.277374e-06
## [17,] -1.357532e-05 -9.010960e-07 -2.495442e-06 -8.426531e-05 -3.788468e-05
## [18,] 2.142286e-07 -6.592187e-07 4.052194e-07 4.825500e-06 1.823939e-07
## [19,] 2.801647e-07 1.157292e-08 4.095682e-08 1.025353e-06 -1.513762e-06
## [20,] 1.050937e-07 1.289198e-07 -2.841969e-08 -5.464631e-07 -6.218474e-07
## [21,] -6.674219e-08 2.462364e-07 -1.092281e-07 -2.596218e-06 -6.546237e-07
## [22,] 1.857110e-08 1.312521e-07 -5.738779e-08 -1.381205e-06 -2.594248e-07
## [23,] 1.495601e-07 1.250464e-07 -1.602544e-08 -6.243374e-08 2.163309e-07
## [24,] 8.612595e-08 2.254942e-07 -3.920547e-08 -3.324771e-07 5.188859e-08
## [25,] -6.935261e-09 1.152495e-07 -1.161922e-08 -1.895544e-07 5.224053e-09
## [26,] -9.429497e-07 -3.023435e-07 1.665569e-07 1.151155e-06 8.683969e-07
## [27,] -1.840622e-06 -4.928702e-07 -3.216630e-07 1.101726e-06 8.638870e-07
## [28,] -3.924879e-07 -1.086537e-06 1.954679e-07 7.519719e-07 5.588906e-07
## [29,] -3.271565e-07 1.900719e-07 -5.902105e-07 -1.573042e-07 -3.994178e-08
## [30,] 6.853742e-07 6.321451e-07 -1.602659e-07 -1.500217e-05 -3.334644e-06
## [31,] 7.560210e-07 3.214569e-07 3.160191e-08 -1.211958e-06 -1.108035e-05
## [32,] 4.054438e-07 3.996381e-08 7.582209e-08 1.978365e-06 -1.970231e-06
## [33,] 5.981705e-07 6.978354e-08 1.577205e-07 5.434308e-06 2.077507e-06
## [34,] 3.799469e-07 1.003813e-07 8.606206e-08 3.188967e-06 1.261034e-06
## [35,] 2.675851e-07 2.229294e-07 2.513920e-10 8.548339e-07 5.989055e-07
## [36,] 2.381560e-07 1.426896e-07 6.713115e-09 7.228871e-07 5.291676e-07
## [37,] 6.438270e-08 2.571688e-08 2.994392e-08 4.696554e-07 3.142671e-07
## [38,] -1.837429e-06 -5.781930e-07 3.385397e-07 2.425151e-06 1.772533e-06
## [39,] -3.495043e-06 -1.266542e-06 -4.089774e-07 3.602710e-06 1.808420e-06
## [40,] -9.530332e-07 -1.309987e-06 3.493619e-08 -1.548019e-06 1.009371e-06
## [41,] -5.259622e-07 -1.244180e-08 -7.354975e-07 1.501045e-06 -4.665577e-08
## [42,] 2.443654e-07 -3.558722e-06 1.946680e-06 4.042431e-05 6.822682e-07
## [43,] 1.277702e-06 -3.149636e-07 3.815724e-07 9.832291e-06 -1.042267e-05
## [44,] 3.583863e-07 3.969837e-07 -8.374967e-08 -1.926610e-06 -3.787341e-06
## [45,] -4.316950e-07 6.922408e-07 -3.968751e-07 -1.028764e-05 -2.828617e-06
## [46,] -6.344467e-08 3.655148e-07 -1.958011e-07 -5.037124e-06 -1.140181e-06
## [47,] 3.857872e-07 3.960385e-07 -4.907719e-08 -8.671168e-08 5.756856e-07
## [48,] 3.312849e-07 3.858658e-07 -4.168412e-08 1.721916e-07 5.398736e-07
## [49,] 5.488035e-08 1.529807e-07 2.391555e-08 2.697005e-07 3.607704e-07
## [50,] -3.956297e-06 -1.235089e-06 7.371284e-07 4.898999e-06 3.601265e-06
## [51,] 3.929439e-04 -2.160261e-06 -8.504239e-07 3.038401e-06 3.542793e-06
## [52,] -2.160261e-06 3.965864e-04 6.021683e-07 7.105261e-06 2.282994e-06
## [53,] -8.504239e-07 6.021683e-07 3.984932e-04 -3.028856e-06 -3.198674e-07
## [54,] 3.038401e-06 7.105261e-06 -3.028856e-06 2.863521e-04 -1.690272e-05
## [55,] 3.542793e-06 2.282994e-06 -3.198674e-07 -1.690272e-05 3.532717e-04
## [56,] 1.492431e-06 1.520648e-07 2.824935e-07 7.133470e-06 -8.824615e-06
## [57,] 2.061324e-06 2.644276e-07 6.086507e-07 2.115522e-05 7.675919e-06
## [58,] 1.229349e-06 4.097710e-07 2.938259e-07 1.131217e-05 4.264992e-06
## [59,] 7.583240e-07 6.830317e-07 -1.409863e-08 2.186001e-06 1.637419e-06
## [60,] 6.123478e-07 5.462364e-07 -2.453321e-08 1.299628e-06 1.225352e-06
## [61,] 1.232347e-07 1.959638e-07 7.428220e-08 1.006008e-06 8.327324e-07
## [62,] -8.281730e-06 -2.576817e-06 1.578102e-06 1.018009e-05 7.418360e-06
## [63,] -1.425069e-05 -5.676894e-06 -7.184947e-07 1.811769e-05 7.931048e-06
## [64,] -4.521949e-06 -3.389323e-06 -3.433861e-07 -1.389849e-05 3.055636e-06
## [65,] -1.453069e-06 -5.865596e-07 -1.735594e-06 1.039450e-05 1.550522e-07
## [66,] -1.594710e-05 3.626311e-05 -8.524880e-06 -1.145423e-04 -4.958512e-05
## [,56] [,57] [,58] [,59] [,60]
## [1,] -1.225304e-06 -5.017787e-06 -4.220555e-06 -2.988305e-06 -2.408010e-06
## [2,] 7.373955e-08 4.354222e-07 2.367368e-07 7.223578e-08 5.281503e-08
## [3,] -6.446816e-07 2.709191e-07 1.470570e-07 4.567180e-08 3.811824e-08
## [4,] -9.707759e-07 -1.173723e-06 -8.086236e-08 -5.569282e-08 -3.150115e-08
## [5,] -8.946744e-07 -3.797297e-06 -1.500183e-06 -1.748094e-07 -6.805816e-08
## [6,] 1.714829e-07 -1.273181e-06 -1.992285e-06 -1.149911e-06 -1.513434e-07
## [7,] 1.269990e-07 1.189169e-07 -1.002151e-06 -1.396583e-06 -3.139658e-07
## [8,] 1.554324e-07 -1.687557e-07 -2.956831e-07 -4.666268e-07 -1.624495e-06
## [9,] 9.427360e-08 -2.337975e-07 -2.035137e-07 -1.623605e-07 -5.618642e-07
## [10,] 1.872756e-07 2.203266e-08 -1.916285e-08 3.285308e-09 1.720695e-08
## [11,] 1.772485e-07 -1.756896e-07 -1.657065e-07 -7.666193e-08 -3.615107e-08
## [12,] 1.768894e-07 2.381085e-07 1.500923e-07 1.031000e-07 8.411383e-08
## [13,] -2.490897e-08 -1.050847e-07 -7.112202e-08 -3.467764e-08 -2.399241e-08
## [14,] -1.779113e-07 5.694711e-06 3.509633e-06 -7.032931e-07 -9.517121e-07
## [15,] -4.984798e-06 -8.865390e-06 -5.811213e-06 -3.223422e-06 -2.566731e-06
## [16,] -2.673772e-06 2.142614e-05 1.051231e-05 -3.899810e-06 -4.279987e-06
## [17,] -1.194017e-05 -2.470218e-05 -1.359724e-05 -4.630767e-06 -3.365368e-06
## [18,] -4.986030e-07 -8.681542e-07 -4.245507e-07 2.690554e-08 4.264741e-08
## [19,] -6.540151e-07 -5.299646e-07 -3.244098e-07 -1.629766e-08 2.007096e-08
## [20,] -5.996019e-08 4.309626e-07 -5.906517e-08 6.385114e-08 6.041758e-08
## [21,] 3.332664e-07 3.222869e-06 2.161859e-06 8.547662e-08 4.712432e-08
## [22,] -1.207121e-07 1.863307e-06 1.537670e-06 -5.457491e-07 -1.083866e-07
## [23,] 7.874036e-08 8.073639e-08 -4.690461e-07 -1.079644e-06 -2.511051e-07
## [24,] 3.409923e-08 3.376781e-07 1.719163e-07 -9.732661e-08 -1.161144e-06
## [25,] 1.058333e-08 2.228266e-07 1.839569e-07 1.035881e-07 -2.143641e-07
## [26,] 2.970636e-07 3.226419e-07 2.004861e-07 1.672219e-07 1.460341e-07
## [27,] 3.337105e-07 3.894039e-07 2.248863e-07 1.500234e-07 1.257446e-07
## [28,] 1.151535e-07 3.152132e-07 2.740403e-07 2.677838e-07 2.056288e-07
## [29,] 4.060101e-08 6.078775e-08 1.530521e-08 -2.796077e-08 -2.246610e-08
## [30,] 7.297109e-07 3.065684e-06 1.684969e-06 4.217451e-07 2.706689e-07
## [31,] -2.367191e-06 1.559209e-06 9.157635e-07 3.867483e-07 2.872345e-07
## [32,] -3.129192e-06 -3.154680e-06 3.298074e-07 1.737232e-07 1.255851e-07
## [33,] -3.194215e-06 -1.199533e-05 -4.119222e-06 3.339195e-07 3.289519e-07
## [34,] 3.500857e-07 -3.934757e-06 -5.315420e-06 -2.426388e-06 -6.662100e-08
## [35,] 2.420886e-07 6.370627e-07 -2.250528e-06 -3.275202e-06 -5.378339e-07
## [36,] 2.000209e-07 4.391676e-07 -1.939050e-08 -5.721107e-07 -3.323526e-06
## [37,] 1.119440e-07 1.425603e-07 8.981100e-08 -1.566859e-08 -9.823947e-07
## [38,] 6.125177e-07 7.785550e-07 4.882032e-07 3.651781e-07 3.054907e-07
## [39,] 5.355741e-07 5.182179e-07 3.265015e-07 3.065681e-07 2.611856e-07
## [40,] 5.770584e-07 1.350364e-06 8.863995e-07 5.034411e-07 3.724199e-07
## [41,] -1.380047e-07 -3.780203e-07 -2.287294e-07 -8.278386e-08 -5.567269e-08
## [42,] -3.923758e-06 -6.986697e-06 -3.635719e-06 -3.320289e-07 -9.991132e-08
## [43,] -3.969371e-06 -2.287844e-06 -1.443921e-06 -6.277747e-08 9.280818e-08
## [44,] 4.333893e-07 2.666710e-06 -2.025274e-07 1.835398e-07 1.831940e-07
## [45,] 2.149683e-06 1.511125e-05 8.682975e-06 -4.109897e-08 -3.930105e-08
## [46,] -4.517727e-07 8.284322e-06 5.936469e-06 -2.129818e-06 -4.150470e-07
## [47,] 2.184847e-07 6.828654e-07 -1.499984e-06 -3.469751e-06 -7.196111e-07
## [48,] 2.134939e-07 6.417429e-07 1.371419e-07 -5.902032e-07 -4.237989e-06
## [49,] 1.350074e-07 3.451225e-07 2.638056e-07 9.950698e-08 -1.053039e-06
## [50,] 1.237830e-06 1.536720e-06 9.634531e-07 7.354653e-07 6.202978e-07
## [51,] 1.492431e-06 2.061324e-06 1.229349e-06 7.583240e-07 6.123478e-07
## [52,] 1.520648e-07 2.644276e-07 4.097710e-07 6.830317e-07 5.462364e-07
## [53,] 2.824935e-07 6.086507e-07 2.938259e-07 -1.409863e-08 -2.453321e-08
## [54,] 7.133470e-06 2.115522e-05 1.131217e-05 2.186001e-06 1.299628e-06
## [55,] -8.824615e-06 7.675919e-06 4.264992e-06 1.637419e-06 1.225352e-06
## [56,] 3.892645e-04 -1.098939e-05 1.159051e-06 6.584598e-07 4.736419e-07
## [57,] -1.098939e-05 3.539496e-04 -1.663391e-05 1.628115e-06 1.350920e-06
## [58,] 1.159051e-06 -1.663391e-05 3.816987e-04 -6.326124e-06 7.029554e-08
## [59,] 6.584598e-07 1.628115e-06 -6.326124e-06 3.908562e-04 -1.379895e-06
## [60,] 4.736419e-07 1.350920e-06 7.029554e-08 -1.379895e-06 3.913899e-04
## [61,] 3.043313e-07 6.346856e-07 4.497465e-07 1.145217e-07 -2.404668e-06
## [62,] 2.550531e-06 3.221291e-06 2.016550e-06 1.514337e-06 1.267976e-06
## [63,] 2.023689e-06 1.944704e-06 1.319331e-06 1.390837e-06 1.174849e-06
## [64,] 2.801987e-06 5.970080e-06 3.650048e-06 1.689025e-06 1.231372e-06
## [65,] -9.390133e-07 -2.094020e-06 -1.162824e-06 -2.845494e-07 -1.805160e-07
## [66,] -1.235441e-05 9.698944e-05 7.743364e-05 4.605987e-05 3.035929e-05
## [,61] [,62] [,63] [,64] [,65]
## [1,] -9.085531e-07 -4.706613e-07 -3.944244e-07 -1.870679e-06 -2.623744e-07
## [2,] 3.640897e-08 3.231449e-07 5.704071e-07 -4.237973e-07 3.281596e-07
## [3,] 3.188060e-08 3.295320e-07 2.721275e-07 2.945506e-07 -1.038818e-07
## [4,] 4.302856e-09 2.645382e-07 2.190975e-07 1.477507e-07 -8.064689e-08
## [5,] 1.141384e-09 4.228842e-07 3.272147e-07 2.671009e-07 -1.633992e-07
## [6,] 1.001888e-08 2.937690e-07 2.270473e-07 2.132269e-07 -1.137058e-07
## [7,] -4.279685e-09 2.454011e-07 2.190376e-07 1.846034e-07 -5.224511e-08
## [8,] -4.881638e-07 2.658591e-07 2.298862e-07 5.532938e-08 -6.269253e-08
## [9,] -3.024492e-07 -1.575453e-07 9.376540e-08 -3.538897e-08 -2.645135e-08
## [10,] -2.900360e-07 -1.871266e-06 -9.292369e-07 -2.251384e-07 7.301313e-08
## [11,] -3.439385e-08 -9.054774e-07 -1.834886e-06 -4.834754e-07 -4.107026e-07
## [12,] 3.891545e-08 -6.092530e-08 -2.615104e-07 -1.001870e-06 1.344665e-07
## [13,] 1.232394e-09 1.262812e-07 -3.498364e-07 1.484420e-07 -5.748620e-07
## [14,] -5.380796e-07 -4.556676e-06 -4.847940e-06 -2.461344e-06 -3.900477e-07
## [15,] -1.278984e-06 -5.646235e-06 -5.345072e-06 -4.269794e-06 3.013515e-07
## [16,] -2.815932e-06 -2.404563e-05 -2.819765e-05 -4.956157e-06 -4.255208e-06
## [17,] -2.423009e-06 -2.317509e-05 -1.902918e-05 -2.326748e-05 7.821312e-06
## [18,] 7.057593e-09 -1.407621e-07 -2.593843e-07 3.030272e-07 -1.506008e-07
## [19,] 3.425739e-08 4.931705e-07 4.419366e-07 3.387291e-07 -1.051856e-07
## [20,] 3.433293e-08 2.453924e-07 2.794607e-07 9.304241e-08 3.257159e-08
## [21,] 1.956202e-08 1.758668e-08 1.741404e-07 -2.830047e-07 2.265220e-07
## [22,] 1.261169e-08 1.143902e-07 2.063853e-07 -1.538405e-07 1.261149e-07
## [23,] 1.395958e-08 3.162738e-07 3.301654e-07 1.829941e-07 -1.009417e-09
## [24,] -2.387025e-07 2.277799e-07 2.360302e-07 2.776452e-07 4.000372e-09
## [25,] -1.273888e-07 -4.363286e-07 3.537076e-08 1.375789e-07 2.015995e-08
## [26,] -4.546384e-07 -3.638440e-06 -1.908258e-06 -3.526062e-07 1.772239e-07
## [27,] 6.225918e-09 -1.855416e-06 -3.518740e-06 -1.014410e-06 -5.788089e-07
## [28,] 7.687360e-08 -4.105649e-07 -1.078868e-06 -1.219132e-06 -3.265516e-08
## [29,] 1.336720e-08 3.240192e-07 -4.182323e-07 1.315142e-08 -7.311689e-07
## [30,] 1.796828e-07 1.572070e-06 2.802106e-06 -2.062801e-06 1.615359e-06
## [31,] 1.791907e-07 1.445762e-06 1.372137e-06 1.076204e-06 -2.147880e-07
## [32,] 8.185267e-08 6.949025e-07 5.560588e-07 7.440060e-07 -2.481778e-07
## [33,] 1.659190e-07 9.620267e-07 6.436762e-07 1.534086e-06 -5.314212e-07
## [34,] 1.246225e-07 6.312407e-07 4.401327e-07 1.004426e-06 -3.222827e-07
## [35,] 3.319196e-08 5.282328e-07 4.794553e-07 5.893129e-07 -1.062475e-07
## [36,] -9.543276e-07 4.706629e-07 4.262831e-07 4.235059e-07 -8.135638e-08
## [37,] -5.712252e-07 -8.098139e-07 1.019899e-07 1.614874e-07 -8.715509e-09
## [38,] -8.968506e-07 -7.530731e-06 -4.139729e-06 -7.535748e-07 3.990821e-07
## [39,] 1.927105e-08 -4.122257e-06 -7.424245e-06 -1.807791e-06 -1.170787e-06
## [40,] 1.550261e-07 -7.109106e-07 -1.736432e-06 -2.889274e-06 3.912310e-07
## [41,] 1.122520e-08 5.095831e-07 -1.052408e-06 4.231532e-07 -1.454784e-06
## [42,] -1.553443e-07 -2.218988e-06 -4.323876e-06 4.207943e-06 -2.775501e-06
## [43,] 1.426685e-07 1.982834e-06 1.346055e-06 2.410581e-06 -1.030463e-06
## [44,] 1.083333e-07 8.183503e-07 9.164065e-07 3.256870e-07 8.559059e-08
## [45,] -1.968500e-08 -3.068736e-07 3.164284e-07 -1.576356e-06 9.297150e-07
## [46,] 6.332221e-09 1.420608e-07 4.685126e-07 -7.379425e-07 4.660659e-07
## [47,] 4.028426e-08 8.261575e-07 8.487383e-07 6.156644e-07 -2.450515e-08
## [48,] -1.059868e-06 7.239407e-07 7.039471e-07 6.697034e-07 -5.605253e-08
## [49,] -7.061983e-07 -1.607439e-06 1.452341e-07 3.136289e-07 2.721466e-08
## [50,] -1.654830e-06 -1.515291e-05 -8.703126e-06 -1.618974e-06 8.884749e-07
## [51,] 1.232347e-07 -8.281730e-06 -1.425069e-05 -4.521949e-06 -1.453069e-06
## [52,] 1.959638e-07 -2.576817e-06 -5.676894e-06 -3.389323e-06 -5.865596e-07
## [53,] 7.428220e-08 1.578102e-06 -7.184947e-07 -3.433861e-07 -1.735594e-06
## [54,] 1.006008e-06 1.018009e-05 1.811769e-05 -1.389849e-05 1.039450e-05
## [55,] 8.327324e-07 7.418360e-06 7.931048e-06 3.055636e-06 1.550522e-07
## [56,] 3.043313e-07 2.550531e-06 2.023689e-06 2.801987e-06 -9.390133e-07
## [57,] 6.346856e-07 3.221291e-06 1.944704e-06 5.970080e-06 -2.094020e-06
## [58,] 4.497465e-07 2.016550e-06 1.319331e-06 3.650048e-06 -1.162824e-06
## [59,] 1.145217e-07 1.514337e-06 1.390837e-06 1.689025e-06 -2.845494e-07
## [60,] -2.404668e-06 1.267976e-06 1.174849e-06 1.231372e-06 -1.805160e-07
## [61,] 3.983830e-04 -3.120332e-06 2.427434e-07 5.708507e-07 1.984397e-08
## [62,] -3.120332e-06 3.690023e-04 -1.860560e-05 -3.505100e-06 1.981864e-06
## [63,] 2.427434e-07 -1.860560e-05 3.689478e-04 -7.583911e-06 -3.460512e-06
## [64,] 5.708507e-07 -3.505100e-06 -7.583911e-06 3.897585e-04 1.665966e-06
## [65,] 1.984397e-08 1.981864e-06 -3.460512e-06 1.665966e-06 3.954011e-04
## [66,] 9.251479e-06 -1.996044e-05 -1.763500e-05 3.001049e-05 4.189432e-06
## [,66]
## [1,] -1.260489e-04
## [2,] -3.626851e-06
## [3,] -6.310892e-06
## [4,] -1.492951e-05
## [5,] -2.623075e-05
## [6,] -1.526080e-05
## [7,] -4.299249e-06
## [8,] -2.218816e-05
## [9,] -1.939079e-05
## [10,] -1.241507e-05
## [11,] -1.998256e-05
## [12,] -1.396871e-06
## [13,] -2.320301e-06
## [14,] 5.702419e-05
## [15,] 8.021419e-05
## [16,] 9.057870e-05
## [17,] 1.570613e-04
## [18,] 1.018219e-05
## [19,] -1.238544e-05
## [20,] 1.297754e-06
## [21,] 7.913297e-06
## [22,] -1.554400e-06
## [23,] 9.980407e-07
## [24,] 1.880316e-05
## [25,] 1.202925e-05
## [26,] -3.571892e-06
## [27,] -6.210078e-06
## [28,] 1.501497e-05
## [29,] -3.646624e-06
## [30,] -8.113671e-06
## [31,] -3.140865e-06
## [32,] -4.359057e-06
## [33,] 1.756284e-05
## [34,] 1.582217e-05
## [35,] 1.449993e-05
## [36,] 4.451148e-06
## [37,] -2.014438e-06
## [38,] -4.765784e-06
## [39,] -8.077459e-06
## [40,] 1.737217e-05
## [41,] -3.046768e-06
## [42,] 5.024874e-05
## [43,] -4.958148e-05
## [44,] 2.047514e-06
## [45,] 9.200804e-06
## [46,] -7.464115e-06
## [47,] 1.460847e-05
## [48,] 2.248193e-05
## [49,] 9.805632e-06
## [50,] -9.925606e-06
## [51,] -1.594710e-05
## [52,] 3.626311e-05
## [53,] -8.524880e-06
## [54,] -1.145423e-04
## [55,] -4.958512e-05
## [56,] -1.235441e-05
## [57,] 9.698944e-05
## [58,] 7.743364e-05
## [59,] 4.605987e-05
## [60,] 3.035929e-05
## [61,] 9.251479e-06
## [62,] -1.996044e-05
## [63,] -1.763500e-05
## [64,] 3.001049e-05
## [65,] 4.189432e-06
## [66,] 5.401626e-03
##
## $log_evidence
## [1] -131.9296
##
## $converge
## [1] "YES"
##
## $iter_counts
## [1] 211
Use the viz_post_coefs() function to visualize
the posterior coefficient summaries for model 3 and model 6, based on
the very strong prior specification.
### add more code chunks if you like
viz_post_coefs(laplace_03_very_strong$mode[-length(laplace_03_very_strong$mode)],
sqrt(diag(laplace_03_very_strong$var_matrix))[-length(laplace_03_very_strong$mode)],
info_03_very_strong$design_matrix %>% colnames())
viz_post_coefs(laplace_06_very_strong$mode[-length(laplace_06_very_strong$mode)],
sqrt(diag(laplace_06_very_strong$var_matrix))[-length(laplace_06_very_strong$mode)],
info_06_very_strong$design_matrix %>% colnames())
Describe the influence of the regression coefficient prior standard deviation on the coefficient posterior distributions.
What do you think?
As the prior becomes stronger, our posterior means become more and more confined around the prior. We can see this in scale of the distribution of the coefficient value become smaller and smaller.
You previously compared the two models using the Bayes Factor based on the weak prior specification.
Compare the performance of the two models with Bayes Factors again, but considering the results based on the strong and very strong priors. Does the prior influence which model is considered to be better?
### add more code chunks if you like
log_evidences <- c(laplace_03_strong$log_evidence, laplace_06_strong$log_evidence)
if (exp(log_evidences[1]) > exp(log_evidences[2])) {
sprintf("Model 3 is better and has bayes factor of:%f", exp(log_evidences[1])/sum(exp(log_evidences)))
} else {
sprintf("Model 6 is better and has bayes factor of:%f", exp(log_evidences[2])/sum(exp(log_evidences)))
}
## [1] "Model 3 is better and has bayes factor of:1.000000"
log_evidences <- c(laplace_03_very_strong$log_evidence, laplace_06_very_strong$log_evidence)
if (exp(log_evidences[1]) > exp(log_evidences[2])) {
sprintf("Model 3 is better and has bayes factor of:%f", exp(log_evidences[1])/sum(exp(log_evidences)))
} else {
sprintf("Model 6 is better and has bayes factor of:%f", exp(log_evidences[2])/sum(exp(log_evidences)))
}
## [1] "Model 6 is better and has bayes factor of:0.999756"
Yes, clearly the prior has influence on the results of the bayes factor.
You examined the behavior of the coefficient posterior based on the influence of the prior. Let’s now consider the prior’s influence by examining the posterior predictive distributions.
You will make posterior predictions following the approach from the previous assignment. Posterior samples are generated and those samples are used to calculate the posterior samples of the mean trend and generate random posterior samples of the response around the mean. In the previous assignment, you made posterior predictions in order to calculate errors. In this assignment, you will not calculate errors, instead you will summarize the posterior predictions of the mean and of the random response.
The generate_lm_post_samples() function is defined for
you below. It uses the MASS::mvrnorm() function generate
posterior samples from the Laplace Approximation’s MVN distribution.
generate_lm_post_samples <- function(mvn_result, length_beta, num_samples)
{
MASS::mvrnorm(n = num_samples,
mu = mvn_result$mode,
Sigma = mvn_result$var_matrix) %>%
as.data.frame() %>% tibble::as_tibble() %>%
purrr::set_names(c(sprintf("beta_%02d", 0:(length_beta-1)), "varphi")) %>%
mutate(sigma = exp(varphi))
}
The code chunk below starts the post_lm_pred_samples()
function. This function generates posterior mean trend predictions and
posterior predictions of the response. The first argument,
Xnew, is a potentially new or test design matrix that we
wish to make predictions at. The second argument, Bmat, is
a matrix of posterior samples of the \(\boldsymbol{\beta}\)-parameters, and the
third argument, sigma_vector, is a vector of posterior
samples of the likelihood noise. The Xnew matrix has rows
equal to the number of predictions points, M, and the
Bmat matrix has rows equal to the number of posterior
samples S.
You must complete the function by performing the necessary matrix
math to calculate the matrix of posterior mean trend predictions,
Umat, and the matrix of posterior response predictions,
Ymat. You must also complete missing arguments to the
definition of the Rmat and Zmat matrices. The
Rmat matrix replicates the posterior likelihood noise
samples the correct number of times. The Zmat matrix is the
matrix of randomly generated standard normal values. You must correctly
specify the required number of rows to the Rmat and
Zmat matrices.
The post_lm_pred_samples() returns the Umat
and Ymat matrices contained within a list.
Perform the necessary matrix math to calculate the matrix of
posterior predicted mean trends Umat and posterior
predicted responses Ymat. You must specify the number of
required rows to create the Rmat and Zmat
matrices.
HINT: The following code chunk should look famaliar…
post_lm_pred_samples <- function(Xnew, Bmat, sigma_vector)
{
# number of new prediction locations
M <- nrow(Xnew)
# number of posterior samples
S <- nrow(Bmat)
# matrix of linear predictors
Umat <- Xnew %*% t(Bmat)
# assmeble matrix of sigma samples, set the number of rows
Rmat <- matrix(rep(sigma_vector, M), M, byrow = TRUE)
# generate standard normal and assemble into matrix
# set the number of rows
Zmat <- matrix(rnorm(M*S), M, byrow = TRUE)
# calculate the random observation predictions
Ymat <- Umat + Rmat * Zmat
# package together
list(Umat = Umat, Ymat = Ymat)
}
Since this assignment is focused on visualizing the predictions, we
will summarize the posterior predictions to focus on the posterior means
and the middle 95% uncertainty intervals. The code chunk below is
defined for you which serves as a useful wrapper function to call
post_lm_pred_samples().
make_post_lm_pred <- function(Xnew, post)
{
Bmat <- post %>% select(starts_with("beta_")) %>% as.matrix()
sigma_vector <- post %>% pull(sigma)
post_lm_pred_samples(Xnew, Bmat, sigma_vector)
}
The code chunk below defines a function
summarize_lm_pred_from_laplace() which manages the actions
necessary to summarize posterior predictions. The first argument,
mvn_result, is the Laplace Approximation object. The second
object is the test design matrix, Xtest, and the third
argument, num_samples, is the number of posterior samples
to make.
You must complete the code chunk below which summarizes the posterior
predictions. This function takes care of most of the coding for you. You
do not have to worry about the generation of the posterior samples OR
calculating the posterior quantiles associated with the middle 95%
uncertainty interval. You must calculate the posterior average by
deciding on whether you should use colMeans() or
rowMeans() to calculate the average across all posterior
samples per prediction location.
Follow the comments in the code chunk below to complete the definition of the summarize_lm_pred_from_laplace() function. You must calculate the average posterior mean trend and the average posterior response.
summarize_lm_pred_from_laplace <- function(mvn_result, Xtest, num_samples)
{
# generate posterior samples of the beta parameters
post <- generate_lm_post_samples(mvn_result, ncol(Xtest), num_samples)
# make posterior predictions on the test set
pred_test <- make_post_lm_pred(Xtest, post)
# calculate summary statistics on the predicted mean and response
# summarize over the posterior samples
# posterior mean, should you summarize along rows (rowMeans) or
# summarize down columns (colMeans) ???
mu_avg <- rowMeans(pred_test$Umat)
y_avg <- rowMeans(pred_test$Ymat)
# posterior quantiles for the middle 95% uncertainty intervals
mu_lwr <- apply(pred_test$Umat, 1, stats::quantile, probs = 0.025)
mu_upr <- apply(pred_test$Umat, 1, stats::quantile, probs = 0.975)
y_lwr <- apply(pred_test$Ymat, 1, stats::quantile, probs = 0.025)
y_upr <- apply(pred_test$Ymat, 1, stats::quantile, probs = 0.975)
# book keeping
tibble::tibble(
mu_avg = mu_avg,
mu_lwr = mu_lwr,
mu_upr = mu_upr,
y_avg = y_avg,
y_lwr = y_lwr,
y_upr = y_upr
) %>%
tibble::rowid_to_column("pred_id")
}
When you made predictions in Problem 02, the lm() object
handled making the test design matrix. However, since we have programmed
the Bayesian modeling approach from scratch we need to create the test
design matrix manually.
Create the test design matrix based on the visualization
grid, viz_grid, using the model 3 formulation. Assign the
result to the X03_test object.
#y ~ (x1 + I(x1^2))*(x2 + I(x2^2)
x03_test <- model.matrix(~ (x1 + I(x1^2))*(x2 + I(x2^2)), data = viz_grid)
Call the summarize_lm_pred_from_laplace()
function to summarize the posterior predictions from the model 3
formulation for the weak, strong, and very strong prior specifications.
Use 5000 posterior samples for each case. Assign the results from the
weak prior to post_pred_summary_viz_03_weak, the results
from the strong prior to post_pred_summary_viz_03_strong,
and the results from the very strong prior to
post_pred_summary_viz_03_very_strong.
### add as many code chunks as you'd like
post_pred_summary_viz_03_very_strong <- summarize_lm_pred_from_laplace(laplace_03_very_strong, x03_test, 5000)
post_pred_summary_viz_03_strong <- summarize_lm_pred_from_laplace(laplace_03_strong, x03_test, 5000)
post_pred_summary_viz_03_weak <- summarize_lm_pred_from_laplace(laplace_03_weak, x03_test, 5000)
You will now visualize the posterior predictions from the model 3
Bayesian models associated with the weak, strong, and very strong
priors. The viz_grid object is joined to the prediction
dataframes assuming you have used the correct variable names!
Visualize the predicted means, confidence intervals, and
prediction intervals in the style of those that you created in Problem
02. The confidence interval bounds are mu_lwr and
mu_upr columns and the prediction interval bounds are the
y_lwr and y_upr columns, respectively. The
posterior predicted mean of the mean is
mu_avg.
Pipe the result of the joined dataframe into
ggplot() and make appropriate aesthetics and layers to
visualize the predictions with the x1 variable mapped to
the x aesthetic and the x2 variable used as a
facet variable.
post_pred_summary_viz_03_weak %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id') %>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = y_lwr, ymax = y_upr), fill = 'orange') +
geom_ribbon(mapping = aes(ymin = mu_lwr, ymax = mu_upr), fill = 'grey') +
geom_line(mapping = aes(y = mu_avg)) +
facet_wrap(~ x2)
post_pred_summary_viz_03_strong %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id') %>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = y_lwr, ymax = y_upr), fill = 'orange') +
geom_ribbon(mapping = aes(ymin = mu_lwr, ymax = mu_upr), fill = 'grey') +
geom_line(mapping = aes(y = mu_avg)) +
facet_wrap(~ x2)
post_pred_summary_viz_03_very_strong %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id') %>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = y_lwr, ymax = y_upr), fill = 'orange') +
geom_ribbon(mapping = aes(ymin = mu_lwr, ymax = mu_upr), fill = 'grey') +
geom_line(mapping = aes(y = mu_avg)) +
facet_wrap(~ x2)
In order to make posterior predictions for the model 6 formulation
you must create a test design matrix consistent with the training set
basis. The code chunk below creates a helper function which extracts the
interior and boundary knots of a natural spline associated with the
training set for you. The first argument, J, is the
degrees-of-freedom (DOF) of the spline, the second argument,
train_data, is the training data set. The third argument
xname is the name of the variable you are applying the
spline to. The xname argument must be
provided as a character string.
make_splines_training_knots <- function(J, train_data, xname)
{
# extract the input from the training set
x <- train_data %>% select(all_of(xname)) %>% pull()
# create the training basis
train_basis <- splines::ns(x, df = J)
# extract the knots
interior_knots <- as.vector(attributes(train_basis)$knots)
boundary_knots <- as.vector(attributes(train_basis)$Boundary.knots)
# book keeping
list(interior_knots = interior_knots,
boundary_knots = boundary_knots)
}
Create the test design matrix based on the visualization
grid, viz_grid, using the model 6 formulation. Assign the
result to the X06_test object. Use the
make_splines_training_knots() function to get the interior
and boundary knots associated with the training set for the
x1 variable to create the test design matrix.
#?all_of
knot_info <- make_splines_training_knots(12, viz_grid, c("x1"))
X06_test = model.matrix(
~ splines::ns(x1,
knots = knot_info$interior_knots,
Boundary.knots = knot_info$boundary_knots) *
(x2 +I(x2^2) + I(x2^3) + I(x2^4)),
data = viz_grid)
Call the summarize_lm_pred_from_laplace()
function to summarize the posterior predictions from the model 6
formulation for the weak, strong, and very strong prior specifications.
Use 5000 posterior samples for each case. Assign the results from the
weak prior to post_pred_summary_viz_06_weak, the results
from the strong prior to post_pred_summary_viz_06_strong,
and the results from the very strong prior to
post_pred_summary_viz_06_very_strong.
HINT: The make_spline_training_knots() function
returns a list! The fields or elements of the list can be accessed via
the $ operator.
### add as many code chunks as you'd like
post_pred_summary_viz_06_weak <- summarize_lm_pred_from_laplace(laplace_06_weak, X06_test, 5000)
post_pred_summary_viz_06_strong <- summarize_lm_pred_from_laplace(laplace_06_strong, X06_test, 5000)
post_pred_summary_viz_06_very_strong <- summarize_lm_pred_from_laplace(laplace_06_very_strong, X06_test, 5000)
You will now visualize the posterior predictions from the model 6
Bayesian models associated with the weak, strong, and very strong
priors. The viz_grid object is joined to the prediction
dataframes assuming you have used the correct variable names!
Visualize the predicted means, confidence intervals, and
prediction intervals in the style of those that you created in Problem
02. The confidence interval bounds are mu_lwr and
mu_upr columns and the prediction interval bounds are the
y_lwr and y_upr columns, respectively. The
posterior predicted mean of the mean is
mu_avg.
Pipe the result of the joined dataframe into
ggplot() and make appropriate aesthetics and layers to
visualize the predictions with the x1 variable mapped to
the x aesthetic and the x2 variable used as a
facet variable.
post_pred_summary_viz_06_weak %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id') %>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = y_lwr, ymax = y_upr), fill = 'orange') +
geom_ribbon(mapping = aes(ymin = mu_lwr, ymax = mu_upr), fill = 'grey') +
geom_line(mapping = aes(y = mu_avg)) +
facet_wrap(~ x2)
post_pred_summary_viz_06_strong %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id') %>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = y_lwr, ymax = y_upr), fill = 'orange') +
geom_ribbon(mapping = aes(ymin = mu_lwr, ymax = mu_upr), fill = 'grey') +
geom_line(mapping = aes(y = mu_avg)) +
facet_wrap(~ x2)
post_pred_summary_viz_06_very_strong %>%
left_join(viz_grid %>% tibble::rowid_to_column("pred_id"),
by = 'pred_id') %>%
ggplot(mapping = aes(x = x1)) +
geom_ribbon(mapping = aes(ymin = y_lwr, ymax = y_upr), fill = 'orange') +
geom_ribbon(mapping = aes(ymin = mu_lwr, ymax = mu_upr), fill = 'grey') +
geom_line(mapping = aes(y = mu_avg)) +
facet_wrap(~ x2)
Describe the behavior of the predictions as the prior standard deviation decreased. Are the posterior predictions consistent with the behavior of the posterior coefficients?
What do you think?
The confidence intervals get tighter with increase in strength of the prior. Yes, as the means are more confined, so is the mean trend of the posterior.
Now that you have worked with Bayesian models with the prior
regularizing the coefficients, you will consider non-Bayesian
regularization methods. You will work with the glmnet
package in this problem which takes care of all fitting and
visualization for you.
The code chunk below loads in glmnet and so you must
have glmnet installed before running this code chunk.
IMPORANT: the eval flag is set to FALSE
below. Once you download glmnet set
eval=TRUE.
library(glmnet)
## Loading required package: Matrix
##
## Attaching package: 'Matrix'
## The following objects are masked from 'package:tidyr':
##
## expand, pack, unpack
## Loaded glmnet 4.1-8
glmnet does not work with the formula interface. And so
you must create the training design matrix. However, glmnet
prefers the the intercept column of ones to not be
included in the design matrix. To support that you must define new
design matrices. These matrices will use the same formulation but you
must remove the intercept column. This is easy to do with the formula
interface and the model.matrix() function. Include
- 1 in the formula and model.matrix() will not
include the intercept. The code chunk below demonstrates removing the
intercept column for a model with linear additive features.
model.matrix( y ~ x1 + x2 - 1, data = df) %>% head()
## x1 x2
## 1 -0.3092328 0.3087799
## 2 0.6312721 -0.5479198
## 3 -0.6827669 2.1664494
## 4 0.2693056 1.2097037
## 5 0.3725202 0.7854860
## 6 1.2966439 -0.1877231
Create the design matrices for glmnet for the
model 3 and model 6 formulations. Remove the intercept column for both
and assign the results to X03_glmnet and
X06_glmnet.
### add more code chunks if you prefer
X03_glmnet <- model.matrix(y ~ (x1 + I(x1^2))*(x2 + I(x2^2) - 1), data = df)
X06_glmnet <- model.matrix(y ~ splines::ns(x1, df = 12) * (x2 +I(x2^2) + I(x2^3) + I(x2^4)) - 1, data = df)
By default glmnet uses the lasso
penalty. Fit a Lasso model by calling glmnet(). The first
argument to glmnet() is the design matrix and the second
argument is a regular vector for the response.
Train a Lasso model for the model 3 and model 6 formulations,
assign the results to lasso_03 and lasso_06,
respectively.
### add more code chunks if you like
#?glmnet
lasso_03 <- glmnet(X03_glmnet, df$y)
lasso_06 <- glmnet(X06_glmnet, df$y)
Plot the coefficient path for each Lasso model by calling the
plot() function on the glmnet model object.
Specify the xvar argument to be 'lambda' in
the plot() call.
### add more code chunks if you like
lasso_03 %>% plot(xvar = 'lambda')
lasso_06 %>% plot(xvar = 'lambda')
Now that you have visualized the coefficient path, it’s time to
identify the best 'lambda' value to use! The
cv.glmnet() function will by default use 10-fold
cross-validation to tune 'lambda'. The first argument to
cv.glmnet() is the design matrix and the second argument is
the regular vector for the response.
Tune the Lasso regularization strength with cross-validation
using the cv.glmnet() function for each model formulation.
Assign the model 3 result to lasso_03_cv_tune and assign
the model 6 result to lasso_06_cv_tune. Also specify the
alpha argument to be 1 to make sure the Lasso penalty is
applied in the cv.glmnet() call.
HINT: The random seed was assigned for you in two separate code chunks below. This will help ensure you can reproduce the cross-validation results.
### the random seed is set for you
set.seed(812312)
### tune the model 3 formulation
lasso_03_cv_tune <- cv.glmnet(X03_glmnet, df$y, alpha = 1)
### the random seed is set for you
set.seed(812312)
### tune the model 6 formulation
lasso_06_cv_tune <- cv.glmnet(X06_glmnet, df$y, alpha = 1)
Plot the cross-validation results using the default plot method for each cross-validation result. How many coefficients are remaining after tuning?
### add more code chunks if you like
lasso_03_cv_tune %>% plot()
lasso_06_cv_tune %>% plot()
Which features have NOT been “turned off” by the Lasso
penalty? Use the coef() function to display the lasso model
cross-validation results to show the tuned penalized regression
coefficients for each model.
Are the final tuned models different from each
other?
### add more code chunks if you like
lasso_03_cv_tune %>% coef()
## 9 x 1 sparse Matrix of class "dgCMatrix"
## s1
## (Intercept) 0.3009999
## x1 .
## I(x1^2) .
## x2 .
## I(x2^2) -0.3040403
## x1:x2 .
## x1:I(x2^2) .
## I(x1^2):x2 .
## I(x1^2):I(x2^2) .
?coef
lasso_06_cv_tune %>% coef()
## 65 x 1 sparse Matrix of class "dgCMatrix"
## s1
## (Intercept) 0.2818858
## splines::ns(x1, df = 12)1 .
## splines::ns(x1, df = 12)2 .
## splines::ns(x1, df = 12)3 .
## splines::ns(x1, df = 12)4 .
## splines::ns(x1, df = 12)5 .
## splines::ns(x1, df = 12)6 .
## splines::ns(x1, df = 12)7 .
## splines::ns(x1, df = 12)8 .
## splines::ns(x1, df = 12)9 .
## splines::ns(x1, df = 12)10 .
## splines::ns(x1, df = 12)11 .
## splines::ns(x1, df = 12)12 .
## x2 .
## I(x2^2) -0.2847331
## I(x2^3) .
## I(x2^4) .
## splines::ns(x1, df = 12)1:x2 .
## splines::ns(x1, df = 12)2:x2 .
## splines::ns(x1, df = 12)3:x2 .
## splines::ns(x1, df = 12)4:x2 .
## splines::ns(x1, df = 12)5:x2 .
## splines::ns(x1, df = 12)6:x2 .
## splines::ns(x1, df = 12)7:x2 .
## splines::ns(x1, df = 12)8:x2 .
## splines::ns(x1, df = 12)9:x2 .
## splines::ns(x1, df = 12)10:x2 .
## splines::ns(x1, df = 12)11:x2 .
## splines::ns(x1, df = 12)12:x2 .
## splines::ns(x1, df = 12)1:I(x2^2) .
## splines::ns(x1, df = 12)2:I(x2^2) .
## splines::ns(x1, df = 12)3:I(x2^2) .
## splines::ns(x1, df = 12)4:I(x2^2) .
## splines::ns(x1, df = 12)5:I(x2^2) .
## splines::ns(x1, df = 12)6:I(x2^2) .
## splines::ns(x1, df = 12)7:I(x2^2) .
## splines::ns(x1, df = 12)8:I(x2^2) .
## splines::ns(x1, df = 12)9:I(x2^2) .
## splines::ns(x1, df = 12)10:I(x2^2) .
## splines::ns(x1, df = 12)11:I(x2^2) .
## splines::ns(x1, df = 12)12:I(x2^2) .
## splines::ns(x1, df = 12)1:I(x2^3) .
## splines::ns(x1, df = 12)2:I(x2^3) .
## splines::ns(x1, df = 12)3:I(x2^3) .
## splines::ns(x1, df = 12)4:I(x2^3) .
## splines::ns(x1, df = 12)5:I(x2^3) .
## splines::ns(x1, df = 12)6:I(x2^3) .
## splines::ns(x1, df = 12)7:I(x2^3) .
## splines::ns(x1, df = 12)8:I(x2^3) .
## splines::ns(x1, df = 12)9:I(x2^3) .
## splines::ns(x1, df = 12)10:I(x2^3) .
## splines::ns(x1, df = 12)11:I(x2^3) .
## splines::ns(x1, df = 12)12:I(x2^3) .
## splines::ns(x1, df = 12)1:I(x2^4) .
## splines::ns(x1, df = 12)2:I(x2^4) .
## splines::ns(x1, df = 12)3:I(x2^4) .
## splines::ns(x1, df = 12)4:I(x2^4) .
## splines::ns(x1, df = 12)5:I(x2^4) .
## splines::ns(x1, df = 12)6:I(x2^4) .
## splines::ns(x1, df = 12)7:I(x2^4) .
## splines::ns(x1, df = 12)8:I(x2^4) .
## splines::ns(x1, df = 12)9:I(x2^4) .
## splines::ns(x1, df = 12)10:I(x2^4) .
## splines::ns(x1, df = 12)11:I(x2^4) .
## splines::ns(x1, df = 12)12:I(x2^4) .